Answer:B 1.5
Step-by-step explanation:
I just did it
Answer:
B 1.5
Step-by-step explanation:
If [infinity] cn8n n = 0 is convergent, can we conclude that each of the following series is convergent? (a) [infinity] cn(−3)n n = 0 When compared to the original series, [infinity] cnxn n = 0 , we see that x = here. Since the original for that particular value of x, we know that this . (b) [infinity] cn(−8)n n = 0 When compared to the original series, [infinity] cnxn n = 0 , we see that x = here. Since the original for that particular value of x, we know that this
Answer:
a) we know that this is convergent.
b) we know that this might not converge.
Step-by-step explanation:
Given the [tex]\sum^\infty_{n=0}C_n8^n[/tex] is convergent
Therefore,
(a) [tex]\sum^\infty_{n=0}C_n(-3)^n[/tex] The power series [tex]\sum C_nx^n[/tex] has radius of convergence at least as big as 8. So we definitely know it converges for all x satisfying -8<x≤8. In particular for x = -3
∴ [tex]\sum^\infty_{n=0}C_n(-3)^n[/tex] is convergent.
(b) [tex]\sum^\infty_{n=0}C_n(-8)^n[/tex] -8 could be right on the edge of the interval of convergence, and so might not converge
The convergence of the series ∑ cn(−3)^n and ∑ cn(−8)^n depends on whether the original power series, ∑ cnxn, converges for these specific values of x i.e. x = -3 and x = -8. To determine this, one must apply the Ratio Test or Root Test.
Explanation:This is a question about the convergence of a series in mathematics, particularly power series. For a power series like ∑ cnxn (from n = 0 to infinity), the series converges absolutely for certain values of x. When dealing with the two series in the question, ∑ cn(−3)^n and ∑ cn(−8)^n, we can observe that they are similar to the original power series, with x = -3 and x = -8, respectively.
Now, whether these series will converge or not, strictly depends on the radius of convergence of the original series. If the original series converges for x = -3 and x = -8, then these two series will also converge. Otherwise, they won't.
To determine the range or radius of convergence, you have to use the Ratio Test or Root Test in most cases. These are some standard mathematical methods used to determine whether a given series is convergent or not.
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Families USA, a monthly magazine that discusses issues related to health and health costs, surveyed 20 of its subscribers. It found that the annual health insurance premiums for a family with coverage through an employer averaged $10,979. The standard deviation of the sample was $1,000.a. Based on this sample information, develop a 90 percent confidence interval for thepopulation mean yearly premium.b. How large a sample is needed to find the population mean within $250 at 99 percentconfidence?
Answer:
(a) The 90 percent confidence interval for the population mean yearly premium is ($10,974.53, $10983.47).
(b) The sample size required is 107.
Step-by-step explanation:
(a)
The (1 - α)% confidence interval for population mean is:
[tex]CI=\bar x\pm t_{\alpha/2, (n-1)}\times \frac{s}{\sqrt{n}}[/tex]
Given:
[tex]\bar x=\$10,979\\s=\$1000\\n=20[/tex]
Compute the critical value of t for 90% confidence level as follows:
[tex]t_{\alpha/2, (n-1)}=t_{0.10/2, (20-1)}=t_{0.05, 19}=1.729[/tex]
*Use a t-table.
Compute the 90% confidence interval for population mean as follows:
[tex]CI=\bar x\pm t_{\alpha/2, (n-1)}\times \frac{s}{\sqrt{n}}[/tex]
[tex]=10979\pm 1.729\times \frac{1000}{\sqrt{20}}\\=10979\pm4.47\\ =(10974.53, 10983.47)[/tex]
Thus, the 90 percent confidence interval for the population mean yearly premium is ($10,974.53, $10983.47).
(b)
The margin of error is provided as:
MOE = $250
The confidence level is, 99%.
The critical value of z for 99% confidence level is:
[tex]z_{\alpha/2}=z_{0.01/2}=z_{0.005}=2.58[/tex]
Compute the sample size as follows:
[tex]MOE= z_{\alpha/2}\times \frac{s}{\sqrt{n}}[/tex]
[tex]n=[\frac{z_{\alpha/2}\times s}{MOE} ]^{2}[/tex]
[tex]=[\frac{2.58\times 1000}{250}]^{2}[/tex]
[tex]=106.5024\\\approx107[/tex]
Thus, the sample size required is 107.
A study of long-distance phone calls made from General Electric's corporate headquarters in Fairfield, Connecticut, revealed the length of the calls, in minutes, follows the normal probability distribution. The mean length of time per call was 3.6 minutes and the standard deviation was 0.40 minutes.
(a) What fraction of the calls last between 3.6 and 4.2 minutes? (Round z-score computation to 2 decimal places and your final answer to 4 decimal places.)
(b) What fraction of the calls last more than 4.2 minutes? (Round z-score computation to 2 decimal places and your final answer to 4 decimal places.)
(c) What fraction of the calls last between 4.2 and 5 minutes? (Round z-score computation to 2 decimal places and final answer to 4 decimal places.)
(d) What fraction of the calls last between 3 and 5 minutes? (Round z-score computation to 2 decimal places and your final answer to 4 decimal places.)
(e) As part of her report to the president, the director of communications would like to report the length of the longest (in duration) 4% of the calls. What is this time? (Round z-score computation to 2 decimal places and your final answer to 2 decimal places.)
Answer:
a) 0.4332 = 43.32% of the calls last between 3.6 and 4.2 minutes
b) 0.0668 = 6.68% of the calls last more than 4.2 minutes
c) 0.0666 = 6.66% of the calls last between 4.2 and 5 minutes
d) 0.9330 = 93.30% of the calls last between 3 and 5 minutes
e) They last at least 4.3 minutes
Step-by-step explanation:
Problems of normally distributed samples are solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
In this problem, we have that:
[tex]\mu = 3.6, \sigma = 0.4[/tex]
(a) What fraction of the calls last between 3.6 and 4.2 minutes?
This is the pvalue of Z when X = 4.2 subtracted by the pvalue of Z when X = 3.6.
X = 4.2
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{4.2 - 3.6}{0.4}[/tex]
[tex]Z = 1.5[/tex]
[tex]Z = 1.5[/tex] has a pvalue of 0.9332
X = 3.6
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{3.6 - 3.6}{0.4}[/tex]
[tex]Z = 0[/tex]
[tex]Z = 0[/tex] has a pvalue of 0.5
0.9332 - 0.5 = 0.4332
0.4332 = 43.32% of the calls last between 3.6 and 4.2 minutes
(b) What fraction of the calls last more than 4.2 minutes?
This is 1 subtracted by the pvalue of Z when X = 4.2. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{4.2 - 3.6}{0.4}[/tex]
[tex]Z = 1.5[/tex]
[tex]Z = 1.5[/tex] has a pvalue of 0.9332
1 - 0.9332 = 0.0668
0.0668 = 6.68% of the calls last more than 4.2 minutes
(c) What fraction of the calls last between 4.2 and 5 minutes?
This is the pvalue of Z when X = 5 subtracted by the pvalue of Z when X = 4.2. So
X = 5
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{5 - 3.6}{0.4}[/tex]
[tex]Z = 3.5[/tex]
[tex]Z = 3.5[/tex] has a pvalue of 0.9998
X = 4.2
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{4.2 - 3.6}{0.4}[/tex]
[tex]Z = 1.5[/tex]
[tex]Z = 1.5[/tex] has a pvalue of 0.9332
0.9998 - 0.9332 = 0.0666
0.0666 = 6.66% of the calls last between 4.2 and 5 minutes
(d) What fraction of the calls last between 3 and 5 minutes?
This is the pvalue of Z when X = 5 subtracted by the pvalue of Z when X = 3.
X = 5
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{5 - 3.6}{0.4}[/tex]
[tex]Z = 3.5[/tex]
[tex]Z = 3.5[/tex] has a pvalue of 0.9998
X = 3
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{3 - 3.6}{0.4}[/tex]
[tex]Z = -1.5[/tex]
[tex]Z = -1.5[/tex] has a pvalue of 0.0668
0.9998 - 0.0668 = 0.9330
0.9330 = 93.30% of the calls last between 3 and 5 minutes
(e) As part of her report to the president, the director of communications would like to report the length of the longest (in duration) 4% of the calls. What is this time?
At least X minutes
X is the 100-4 = 96th percentile, which is found when Z has a pvalue of 0.96. So X when Z = 1.75.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]1.75 = \frac{X - 3.6}{0.4}[/tex]
[tex]X - 3.6 = 0.4*1.75[/tex]
[tex]X = 4.3[/tex]
They last at least 4.3 minutes
A computer manufacturer is testing a batch of processors. They place a simple random sample of processors from the batch under a stress test and record the number of failures, their guidelines specify the percentage of failures should be under 4%. Of the 300 processors tested, there were 54 failures.
a. State the hypotheses for the test. Assume the manufacturer wants to assume there is a problem with a batch, that is, they will only accept that there are fewer than 4% failures in the population if they have evidence for it.
b. Calculate the test statistic and p-value. Your test statistic should be either a z-value or a t-value, whichever is appropriate for the problem
Answer:
a)We need to conduct a hypothesis in order to test the claim that the true proportion is lower than 0.04 or no.:
Null hypothesis:[tex]p \leq 0.04[/tex]
Alternative hypothesis:[tex]p > 0.04[/tex]
When we conduct a proportion test we need to use the z statistic, and the is given by:
b) We need to use a z statistic
Since we have all the info requires we can replace in formula (1) like this:
[tex]z=\frac{0.135 -0.04}{\sqrt{\frac{0.04(1-0.04)}{300}}}=8.396[/tex]
Since is a right tailed test the p value would be:
[tex]p_v =P(z>8.396) \approx 0[/tex]
Step-by-step explanation:
Data given and notation
n=400 represent the random sample taken
X=54 represent the number of failures
[tex]\hat p=\frac{54}{400}=0.135[/tex] estimated proportion of adults that said that it is morally wrong to not report all income on tax returns
[tex]p_o=0.04[/tex] is the value that we want to test
[tex]\alpha[/tex] represent the significance level
z would represent the statistic (variable of interest)
[tex]p_v[/tex] represent the p value (variable of interest)
Part a: Concepts and formulas to use
We need to conduct a hypothesis in order to test the claim that the true proportion is lower than 0.04 or no.:
Null hypothesis:[tex]p \leq 0.04[/tex]
Alternative hypothesis:[tex]p > 0.04[/tex]
When we conduct a proportion test we need to use the z statistic, and the is given by:
[tex]z=\frac{\hat p -p_o}{\sqrt{\frac{p_o (1-p_o)}{n}}}[/tex] (1)
The One-Sample Proportion Test is used to assess whether a population proportion [tex]\hat p[/tex] is significantly different from a hypothesized value [tex]p_o[/tex].
Part b: Calculate the statistic
We need to use a z statistic
Since we have all the info requires we can replace in formula (1) like this:
[tex]z=\frac{0.135 -0.04}{\sqrt{\frac{0.04(1-0.04)}{300}}}=8.396[/tex]
Statistical decision
It's important to refresh the p value method or p value approach . "This method is about determining "likely" or "unlikely" by determining the probability assuming the null hypothesis were true of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed". Or in other words is just a method to have an statistical decision to fail to reject or reject the null hypothesis.
The significance level provided [tex]\alpha=0.05[/tex]. The next step would be calculate the p value for this test.
Since is a right tailed test the p value would be:
[tex]p_v =P(z>8.396) \approx 0[/tex]
So the p value obtained was a very low value and using the significance level for example [tex]\alpha=0.05[/tex] we have [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, and we can said that at 5% of significance the proportion of defectives is significantly higher than 0.04 or 4%
a study examined the effectiveness of an ear molding technique used to correct ear deformities in newborns without the need for surgery. a plastic mold was placed around the deformed ear shortly after birth for about 2 weeks. the study reports that 152 of 158 procedures were successful. what is the sample proportion of successful procedures
Answer:
The sample proportion of successful procedures is 0.962.
Step-by-step explanation:
The sample proportion of successful procedures is the number of successfiç procedures divided by the total number of procedures.
In this problem:
158 procedures, of which 152 were successful. So
p = 152/158 = 0.962
The sample proportion of successful procedures is 0.962.
The sample proportion is the ratio of the number of successes to the total number of samples or trials, Hence, the sample proportion is 0.962
Given the Parameters :
Number of successful procedures, x = 152 Total number of procedures, n = 158The sample proportion can be calculated using the relation :
Sample proportion = x/nSample proportion = 152/158 = 0.962
Hence, the sampling proportion of successful procedures is 0.962.
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A study was recently conducted to estimate the mean cholesterol for adult males over the age of 55 years. The following random sample data were observed: 245 304 135 202 300 196 210 188 256 390 Given this information, what is the point estimate for the population mean? Question 2 options: About 242.6 ±102 About 73.35 Can't be determined without knowing the confidence level.
Final answer:
The point estimate for the population mean cholesterol level for adult males over the age of 55 years is 242.6 mg/dL, calculated by averaging the values from the given sample data.
Explanation:
The point estimate for the population mean is the average of the sample data. Here are the steps to calculate it:
Add all the sample values together to get the sum: 245 + 304 + 135 + 202 + 300 + 196 + 210 + 188 + 256 + 390 = 2,426.Divide the sum by the number of data points to find the mean: 2,426 / 10 = 242.6.So, the point estimate for the population mean cholesterol level for adult males over the age of 55 years is 242.6 mg/dL. This value is the best estimate for the mean cholesterol level based on the sample given.
What is the approximate length of the missing side in the triangle below?
18 miles
15 miles
80 degrees
[Not drawn to scale]
Answer: 21.3 miles
Step-by-step explanation:
Find the product
(9x2 – 4).
An expression can be written in rational form by
writing it as a fraction with a denominator
of |
Answer:
14/1
Step-by-step explanation:
Answer:
The answer is 1. The next part of that same question should be 12n+8.
Find X: 5x=45
A) 9
B) 2
C) 12
D) 0
Answer: [tex]x=9[/tex]
Divide both side by 5
[tex]5x/5=45/5\\x=9[/tex]
Answer:
[tex]x = 9[/tex]
Step-by-step explanation:
[tex]5x = 45 \\ \frac{5x}{5} = \frac{45}{5} \\ x = 9[/tex]
hope this helps you.....
Peanuts cost $12.00 for 2.5 pounds.How much for 1 pound
Answer:
$4.80
Step-by-step explanation:
Make a proportion
$12 for 2.5 pounds, and $x for 1 pound
12/2.5=x/1
x/1 is equivalent to x
12/2.5=x
Divide
x=4.8
So, one pound of peanuts costs $4.80
Answer:
4.8
Step-by-step explanation:
divide 12.00 by 2.5 to find the unit rate which is 4.8.
which of these expressions is equivalent to log (3 x 8)
Answer:log3 + log8
Step-by-step explanation:
log(3x8)=log3 + log8
The sum of number times 3 and 15
Answer:
45?
Step-by-step explanation:
How would you not know this?
Which statements are true about the shapes? Select three options.
Figure A is a cylinder. Figure B is a cone. Figure C is a sphere. Figure D is a pyramid with rectangular base.
Figure A is a cylinder.
Figure B is a square pyramid.
Figure C has no bases.
Figure D is a triangular prism.
Figure D has four lateral faces that are triangles
Correctly identifying shapes involves recognizing defining features. Figures A, C, and E are accurately labeled, while B and D need correction due to misclassifications.
In mathematics, shapes are defined by their boundaries or contours, often enclosed by points, lines, curves, and more. These characteristics categorize shapes into various types. The identification of shapes involves recognizing their defining features.
Analyzing the provided shapes:
Figure A is correctly identified as a cylinder. Its appearance aligns with the characteristics of a cylinder.
Figure B is mistakenly labeled as a square pyramid when, in fact, it resembles a cone. This discrepancy points to an incorrect classification.
Figure C is accurately described as having no bases, resembling a sphere. The absence of a base is a defining feature of a sphere.
Figure D is erroneously labeled as a triangular prism, whereas it more closely resembles a rectangular prism. This misclassification may lead to confusion.
Additionally, Figure D is correctly recognized for having four lateral faces that are triangles, aligning with the characteristics of a rectangular prism.
In summary, the accurate identifications are A, C, and E, while B and D require correction based on their actual geometric features.
A real estate appraiser is developing a regression model to predict the market value of single family residential houses as a function of heated area, number of bedrooms, number of bathrooms, age of the house, and central heating (yes, no). The response variable in this model is _______.
a. heated area
b. number of bedrooms
c. market value
d. central heating
e. residential houses
The response variable in this model is: Market value .
Hence the correct option is C.
Given, that a regression model is developed.
A real estate appraiser is developing a regression model to predict the market value of single family residential houses as a function of heated area, number of bedrooms, number of bathrooms, age of the house, and central heating (yes, no). The response variable in this model is market value .
Therefore the correct option is C .
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In the regression model being developed by the real estate appraiser, the response variable is the market value of the single-family residential houses.
Explanation:In the regression model described by the question, the market value of single-family residential houses is being predicted based on several predictive variables (heated area, number of bedrooms, number of bathrooms, age of the house, and presence of central heating). Therefore, the response variable or the dependent variable in this model is the market value of the houses.
A response variable is the feature or quantity that the model outputs. In this context, the real estate appraiser is trying to produce a model that will tell him how changes in the variables like heated area, bedrooms etc. will change the output, which is the market value.
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A survey reported in Time magazine included the question ‘‘Do you favor a federal law requiring a 15 day waiting period to purchase a gun?" Results from a random sample of US citizens showed that 318 of the 520 men who were surveyed supported this proposed law while 379 of the 460 women sampled said ‘‘yes". Use this information to find a 95% confidence interval for the difference in the two proportions, , where is the proportion of men who support the proposed law and is the proportion of women who support the proposed law. Round your answers to three decimal places. The 95% confidence interval is Enter your answer; The 95%confidence interval, value 1 to Enter your answer; The 95%confidence interval, value 2 .
Answer:
Step-by-step explanation:
Confidence interval for the difference in the two proportions is written as
Difference in sample proportions ± margin of error
Sample proportion, p= x/n
Where x = number of success
n = number of samples
For the men,
x = 318
n1 = 520
p1 = 318/520 = 0.61
For the women
x = 379
n2 = 460
p2 = 379/460 = 0.82
Margin of error = z√[p1(1 - p1)/n1 + p2(1 - p2)/n2]
To determine the z score, we subtract the confidence level from 100% to get α
α = 1 - 0.95 = 0.05
α/2 = 0.05/2 = 0.025
This is the area in each tail. Since we want the area in the middle, it becomes
1 - 0.025 = 0.975
The z score corresponding to the area on the z table is 1.96. Thus, confidence level of 95% is 1.96
Margin of error = 1.96 × √[0.61(1 - 0.61)/520 + 0.82(1 - 0.82)/460]
= 1.96 × √0.0004575 + 0.00032086957)
= 0.055
Confidence interval = 0.61 - 0.82 ± 0.055
= - 0.21 ± 0.055
Factor x3+x2+x+1 by grouping. What is the resulting expression
Answer:
(x+1) (x^2+1)
Step-by-step explanation:
x^3+x^2+x+1
Factor by grouping
x^3+x^2 +x+1
Factor out x^2 from the first group and 1 from the second group
x^2( x+1) + 1( x+1)
Factor out (x+1)
(x+1) (x^2+1)
The temperature at a point (x, y) on a flat metal plate is given by T(x, y) = 31/(3 + x^2 + y^2), where T is measured in °C and x, y in meters.
(a) Find the rate of change of temperature with respect to distance at the point (3, 9) in the x- direction.
(b) Find the rate of change of temperature with respect to distance at the point (3, 9) in the y- direction.
Answer:
Find the rate of change of temperature with respect to distance at the point (3, 9) in the x- direction.
(b) Find the rate of change of temperature with respect to distance at the point (3, 9) in the y-
Step-by-step explanation:
because 3 9 t 2 x y where t
Dr. Cawood's class is presented with a large bin that contains small plastic beads. The beads in the box are either purple or white. Dr. Cawood states that 25% of the beads are purple but Oliver, a student in Dr. Cawood's class believes that more than a quarter of the beads in the model box are purple. Select both a null and alternative hypothesis that represent this situation. Be careful of notation.
Answer:
Step-by-step explanation:
The null hypothesis is the hypothesis that is assumed to be true. It is an expression that is the opposite of what the researcher predicts.
The alternative hypothesis is what the researcher expects or predicts. It is the statement that is believed to be true if the null hypothesis is rejected.
From the given situation,
Dr. Cawood states that 25% of the beads are purple. This is the null hypothesis.
Oliver, a student in Dr. Cawood's class believes that more than a quarter of the beads in the model box are purple. This is the alternative hypothesis.
Therefore, the correct null and alternative hypotheses are
H0: p = 0.25 and HA: p > 0.25
The null hypothesis, [tex]\(H_0\)[/tex], is that the proportion of purple beads in the bin is 25%, which can be written as [tex]\(H_0: p = 0.25\)[/tex]. The alternative hypothesis, [tex]\(H_1\)[/tex], is that the proportion of purple beads is greater than 25%, which can be written as [tex]\(H_1: p > 0.25\)[/tex].
In hypothesis testing, the null hypothesis represents the default position or the status quo, which in this case is Dr. Cawood's claim that 25% of the beads are purple. This is represented by the symbol [tex]\(H_0\)[/tex] and is often a statement of no effect or no difference. Here, it is stated as [tex]\(H_0: p = 0.25\)[/tex], where p is the proportion of purple beads in the bin.
The alternative hypothesis, denoted by [tex]\(H_1\)[/tex], represents the claim that is being tested against the null hypothesis. It is the hypothesis that the researcher or the student, in this case Oliver, believes to be true. Since Oliver believes that more than a quarter of the beads are purple, the alternative hypothesis is that the proportion of purple beads is greater than 25%. This is represented as [tex]\(H_1: p > 0.25\)[/tex].
It is important to note that the alternative hypothesis can also be one-sided (as in this case) or two-sided, depending on the context of the claim being tested. If Oliver believed that the proportion of purple beads was different from 25% (either more or less), the alternative hypothesis would be [tex]\(H_1: p \neq 0.25\)[/tex], indicating a two-sided test. However, since the claim specifies ""more than,"" a one-sided alternative hypothesis is appropriate.
A random sample of 16 students selected from the student body of a large university had an average age of 25 years. We want to determine if the average age of all the students at the university is significantly different from 24. Assume the distribution of the population of ages is normal with a standard deviation of 2 years. At a .05 level of significance, it can be concluded that the mean age is:
Answer:
[tex]z=\frac{25-24}{\frac{2}{\sqrt{16}}}=2[/tex]
[tex]p_v =2*P(Z>2)=0.0455[/tex]
If we compare the p value and the significance level given [tex]\alpha=0.05[/tex] we see that [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, and we can conclude that the true mean differs from 24 at 5% of significance
Step-by-step explanation:
Data given and notation
[tex]\bar X=25[/tex] represent the sample mean
[tex]\sigma=2[/tex] represent the sample population deviation for the sample
[tex]n=16[/tex] sample size
[tex]\mu_o =24[/tex] represent the value that we want to test
[tex]\alpha=0.05[/tex] represent the significance level for the hypothesis test.
t would represent the statistic (variable of interest)
[tex]p_v[/tex] represent the p value for the test (variable of interest)
State the null and alternative hypotheses.
We need to conduct a hypothesis in order to check if the true mean is different from 24, the system of hypothesis would be:
Null hypothesis:[tex]\mu = 24[/tex]
Alternative hypothesis:[tex]\mu \neq 24[/tex]
If we analyze the size for the sample is < 30 but we know the population deviation so is better apply a z test to compare the actual mean to the reference value, and the statistic is given by:
[tex]z=\frac{\bar X-\mu_o}{\frac{\sigma}{\sqrt{n}}}[/tex] (1)
z-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".
Calculate the statistic
We can replace in formula (1) the info given like this:
[tex]z=\frac{25-24}{\frac{2}{\sqrt{16}}}=2[/tex]
P-value
Since is a two sided test the p value would be:
[tex]p_v =2*P(Z>2)=0.0455[/tex]
Conclusion
If we compare the p value and the significance level given [tex]\alpha=0.05[/tex] we see that [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, and we can conclude that the true mean differs from 24 at 5% of significance
A hypothesis test is used to determine if the average age of all the students at the university is significantly different from 24. The test statistic does not fall within the critical region, so we fail to reject the null hypothesis. There is not enough evidence to conclude that the mean age is significantly different from 24.
Explanation:In order to determine if the average age of all the students at the university is significantly different from 24, we can perform a hypothesis test.
Step 1: State the hypotheses:
Null hypothesis (H0): The mean age is 24.Alternate hypothesis (Ha): The mean age is not equal to 24.Step 2: Set the significance level (α): α = 0.05.
Step 3: Calculate the test statistic:
Standard error (SE) = standard deviation / √sample size = 2 / √16 = 2 / 4 = 0.5.T-test statistic = (sample mean - population mean) / SE = (25 - 24) / 0.5 = 2.Step 4: Determine the critical value(s): Since the test statistic follows a t-distribution, we need to find the critical values from the t-table. With a sample size of 16 and a significance level of 0.05, we have 15 degrees of freedom. The critical values are t = ±2.131.
Step 5: Make a decision: Since the test statistic (2) does not fall within the critical region (±2.131), we fail to reject the null hypothesis. There is not enough evidence to conclude that the mean age is significantly different from 24.
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The exponential model Upper A equals 925.2 e Superscript 0.027 t describes the population, A, of a country in millions, t years after 2003. Use the model to determine when the population of the country will be 1504 million.
Answer:
In 2021 , the population of the country will be 1504 million
Step-by-step explanation:
We are given that
[tex]A=925.2e^{0.027t}[/tex]
Where A(in millions)
Time,t=After 2003
We have to find the population of the country will be 1504 million.
Substitute the values
[tex]1504=925.2e^{0.027t}[/tex]
[tex]e^{0.027t}=\frac{1504}{925.2}[/tex]
[tex]e^{0.027t}=1.626[/tex]
[tex]0.027t=ln(1.626)[/tex]
[tex]0.027t=0.486[/tex]
[tex]t=\frac{0.486}{0.027}[/tex]
[tex]t=18[/tex]
After 18 years means=2003+18=2021
In 2021 , the population of country will be 1504 million.
what are some of the characteristics of the graph of a cube root functions
Answer:
I)One to three roots.
ii)Two or zero extrema.
iii)One inflection point.
iv)Point symmetry about the inflection point.
v)Range is the set of real numbers.
vi)Three fundamental shapes.
vii)Four points or pieces of information are required to define a cubic polynomial function.
Roots are solvable by radicals.
Final answer:
The graph of a cube root function has several characteristics such as its domain, range, starting point, and symmetry.
Explanation:
A cube root function is represented by the equation y = ∛x. The graph of a cube root function has several characteristics:
The domain of the function is all real numbers because the cube root can be taken of any number.The range of the function is also all real numbers because the cube root of any number will give a real number.The graph starts in the fourth quadrant and passes through the origin.The graph is continuous, meaning there are no breaks or holes in the line.The graph is decreasing for negative values of x and increasing for positive values of x.The graph is symmetric about the origin, meaning if a point (x, y) lies on the graph, then (-x, -y) also lies on the graph.According to an IRS study, it takes a mean of 330 minutes for taxpayers to prepare, copy, and electronically file a 1040 tax form. This distribution of times follows the normal distribution and the standard deviation is 80 minutes. A consumer watchdog agency selects a random sample of 40 taxpayers. What is the likelihood the sample mean is greater than 320 minutes?
Answer:
78.52% probability that the sample mean is greater than 320 minutes
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal probability distribution
Problems of normally distributed samples are solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
In this problem, we have that:
[tex]\mu = 330, \sigma = 80, n = 40, s = \frac{80}{\sqrt{40}} = 12.65[/tex]
What is the likelihood the sample mean is greater than 320 minutes?
This is 1 subtracted by the pvalue of Z when X = 320. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{320 - 330}{12.65}[/tex]
[tex]Z = -0.79[/tex]
[tex]Z = -0.79[/tex] has a pvalue of 0.2148
1 - 0.2148 = 0.7852
78.52% probability that the sample mean is greater than 320 minutes
A rectangle has a height of 3c4 and a width of c2– 4c + 3.
Express the area of the entire rectangle.
Your answer should be a polynomial in standard form.
+
-4c
+
3
3c4
Answer: 3c^6 - 12c^5 + 9c^4
Step-by-step explanation:
The area of the rectangle is found by multiplying the height (3c^4) and the width (c^2 - 4c + 3), resulting in 3c^6 - 12c^5 + 9c^4.
Explanation:To find the area of a rectangle, you use the formula Area = length x width. From your question, we are told that the height of the rectangle is 3c4, and the width is c2 - 4c + 3. So, to find the area, we need to multiply the height and the width. That will give us:
Area = 3c4(c2 - 4c + 3)
Which, upon multiplication, yields Area = 3c6 - 12c5 + 9c4.
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A simple random sample of 450 residents in the state of New York is taken to estimate the proportion of people who live within one mile of a hazardous waste site. If 135 of the residents in the sample live within one mile of a hazardous waste site, what are the values of the sample proportion of people who live within one mile of a hazardous waste site and its standard error?
A. 0.3 and 0.01
B. 0.3 and 0.022
C. 0.3 and 0.21
D. 0.7 and 0.022
Answer:
B. 0.3 and 0.022
Step-by-step explanation:
The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
For a sample proportion p in a sample of size n, the standard error is [tex]s = \sqrt{\frac{p(1-p)}{n}}[/tex]
135 of the 450 residents sampled live within one mile of a hazardous waste site.
So the sample proportion is [tex]p = \frac{135}{450} = 0.3[/tex]
Standard error
[tex]s = \sqrt{\frac{0.3*0.7}{450}} = 0.022[/tex]
So the correct answer is:
B. 0.3 and 0.022
Final answer:
The sample proportion of people who live within one mile of a hazardous waste site is 0.3, and the standard error for this sample proportion is 0.022. The correct answer is B. 0.3 and 0.022.
Explanation:
To estimate the proportion of people who live within one mile of a hazardous waste site using a sample, we divide the number of residents living within one mile by the total number of residents sampled. In this case, 135 residents live within one mile out of a sample of 450 residents, which makes the sample proportion 135/450 = 0.3.
To calculate the standard error (SE) for the sample proportion, we use the formula SE = √(p(1-p)/n), where p is the sample proportion and n is the sample size. Plugging in the values, we get SE = √(0.3(1-0.3)/450), which approximates to 0.022.
The correct answer is B. 0.3 and 0.022 for the sample proportion and its standard error, respectively.
Each marble bag sold by Hans's Marble Company contains 8 purple marbles for every 5 green marbles. If a bag has 30 green marbles, how many purple marbles does it contain?
We can set up a proportion since we know the ratio between the # of purple marbles and the # of green marbles.
A ratio is a way to compare two different values: for every 8 purple marbles there are 5 green marbles
[tex]\frac{8}{5} = \frac{p}{30}[/tex]when we cross multiply, we get 5p = 240 we divide both sides by 5 to isolate the variable There are 48 purple marbles
What is the answer ?
Answer:
its d
Step-by-step explanation:
Kathy and her brother Clay recently ran in a local marathon. The distribution of finishing time for women was approximately normal with mean 259 minutes and standard deviation 32 minutes. The distribution of finishing time for men was approximately normal with mean 242 minutes and standard deviation 29 minutes.
(a) The finishing time for Clay was 289 minutes. Calculate and interpret the standardized score for Clay’s marathon time. Show your work.
(b) The finishing time for Kathy was 272 minutes. What proportion of women who ran the marathon had a finishing time less than Kathy’s? Show your work.
(c) The standard deviation of finishing time is greater for women than for men. What does this indicate about the finishing times of the women who ran the marathon compared to the finishing times of the men who ran the marathon?
Answer:
a) Clay's marathon time is 1.62 standard deviations above the mean finishing time for men.
b) 69.51% of those who ran the marathon has a finishing time less than 272.
c) The spread in the distribution of women's finishing time is greater as compared to the spread in the distribution of men's finishing time.
Step-by-step explanation:
Part a) It was given that, the finishing time for Clay's marathon time is 289 minutes.
To calculate the standardized test score for Clay's marathon time, we use the formula:
[tex]z = \frac{x - \mu}{ \sigma} [/tex]
where
[tex]x = 289[/tex]
[tex] \mu = 242[/tex]
and
[tex] \sigma = 29[/tex]
We substitute the values into the formula to get:
[tex]z = \frac{289 - 242}{29} = 1.62[/tex]
Interpretation: Clay's marathon time is 1.62 standard deviations above the mean finishing time for men.
b) To calculate the proportion of women that had a finishing time less than Kathy , we again need to calculate the z-score for x=272, with mean for women being 259 minutes and standard deviation 32 minutes.
We substitute to get:
[tex]z = \frac{272 - 259}{32} = 0.41[/tex]
From the standard normal distribution table, P(z<0.41)=0.6951
Therefore 69.51% of those who ran the marathon has a finishing time less than 272.
c) The standard deviation measures the variation of a distribution. This means the standard deviation measures how far away the data set of a distribution are from the mean.
If the standard deviation of finishing time is greater for women than for men, then it indicates that, women's finishing time are far away from the mean finishing time as compared to men's finishing time.
Is the square root of 95 less than 10
Answer:
Yes,it is
Step-by-step explanation:
95's square root is 9.7467943448
10 is greater than 9.7 and so forth.
So the square root of 95 is less than 10.
The square root of 95 is approximately 9.74679434, thus the square root of 95 is less than 10. The question pertains to the mathematical operation of finding square roots that are indeed pivotal while solving various mathematical problems.
The student question asks if the square root of 95 is less than 10. The square root of a number is a value that, when multiplied by itself, gives the original number. In this case, the square root of 95 is approximately 9.74679434, which is indeed less than 10. The concept of square roots comes from the realm of Mathematics, more specifically Algebra. It's crucial to understand this mathematical operation as it is frequently encountered in various mathematical problems, especially ones involving quadratic equations where an unknown variable is squared. Usually, these equations will yield two solutions, as both a positive and a negative number squared gives the same result. However, the context of a problem can sometimes restrict the solution to only one value, typically the positive.
As an example, let's consider an equation like x² = 49. The solutions to this are x = -7 and x = 7 because both (-7)² and 7² equals 49. But if this equation was describing a real-world scenario where negative values could not apply (like time or distance), the meaningful solution would only be x = 7.
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find the value of x and tan x in the given trigonometric equation:
sin 2x= cos(3x+10)
Answer:
x = 16°, tan x = 0.3
Step-by-step explanation:
[tex]sin \: 2x = cos \: (3x + 10) \\ cos(90 - 2x) = cos \: (3x + 10) \\ 90 - 2x = 3x + 10 \\ 90 - 10 = 3x + 2x \\ 80 = 5x \\ x = \frac{80}{5} \\ \huge \red{ \boxed{x = 16 \degree }}\\ \\ tan \: x = tan \: 16 \degree \\ = 0.2867453858 \\ = 0.3[/tex]
A hypnosis program designed to help individuals quit smoking claims a 57% success rate. A consumer watch group suspects that this claim is high and randomly selects 50 individuals who have completed the program in order to conduct a hypothesis test. What are the appropriate null and alternative hypotheses?
a) H0: p = 0.57 vs. Ha: p < 0.57
b) H0: p = 0.57 vs. Ha: p > 0.57
c) H0: p = 0.57 vs. Ha: p ≠ 0.57
d) H0: p < 0.57 vs. Ha: p > 0.57
The appropriate null and alternative hypotheses for the consumer watch group's hypothesis test on the hypnosis program's success rate are:
option a) H0: p = 0.57 vs. Ha: p < 0.57
The Null Hypothesis (H0):The success rate of the hypnosis program is 57%.
Alternative Hypothesis (Ha): The success rate is less than 57%.
We use a one-tailed test because we have a directional hypothesis (suspecting a lower success rate).
If we suspected a higher or different rate, we'd use a two-tailed test.
The consumer watch group is challenging the claim of a 57% success rate by testing if the actual success rate is lower than 57%. This setup allows for a focused investigation into the program's effectiveness.
The complete question is :A hypnosis program designed to help individuals quit smoking claims a 57% success rate. A consumer watch group suspects that this claim is high and randomly selects 50 individuals who have completed the program in order to conduct a hypothesis test. What are the appropriate null and alternative hypotheses?
a) H0: p = 0.57 vs. Ha: p < 0.57
b) H0: p = 0.57 vs. Ha: p > 0.57
c) H0: p = 0.57 vs. Ha: p ≠ 0.57
d) H0: p < 0.57 vs. Ha: p > 0.57