Question 1

Under what three conditions is it appropriate to use the t distribution in place of the standard normal distribution?

Accepted Answer

1) $\mathrm { n } \leq 30$
2) $\sigma$ is unknown, and
3) the parent population is essentially normal

Question 2

Describe the process for finding the confidence interval for a population proportion.

Accepted Answer

1) Find the summary statistics $\hat { \mathrm { p } }$ and $\hat { \mathrm { q } }$
2) Compute E using the z distribution and the formula
$E = { } _ { z } \alpha / 2 \cdot \sqrt { \frac { \hat { p q } } { \mathrm { n } } } .$
3) Find the interval by adding $E$ to $\hat { p }$ and then subtracting $E$ from $\hat { p }$.
4) Interpret the interval.

Question 3

Express the confidence interval using the indicated format.
-Express the confidence interval $- 0.061 < \mathrm { p } < 0.579$ in the form of $\hat { \mathrm { p } } \pm \mathrm { E }$.

Accepted Answer

A)  $0.259 \pm 0.32$ 
B)  $0.259 - 0.32$ 
C)  $0.259 \pm 0.5$ 
D)  $0.32 \pm 0.5$ 
A)  $0.259 \pm 0.32$ 
B)  $0.259 - 0.32$ 
C)  $0.259 \pm 0.5$ 
D)  $0.32 \pm 0.5$ A

Question 4

The following confidence interval is obtained for a population proportion, $p : ( 0.734,0.778 )$ Use these confidence interval limits to find the margin of error, E.

Accepted Answer

A)  0.020 
B)  0.044 
C)  0.022 
D)  0.023 
A)  0.020 
B)  0.044 
C)  0.022 
D)  0.023

Question 5

Hannah selected a simple random sample of all adults in her town and, based on this sample, constructed a confidence interval for the mean salary of all adults in the town.
However, the distribution of salaries in the town is not exactly normal. Will the confidence interval still give a good estimate of the mean salary?

Accepted Answer

Yes, confidence interval procedures are

Question 6

Use the given degree of confidence and sample data to find a confidence interval for the population standard deviation σ.
Assume that the population has a normal distribution. Round the confidence interval limits to the same number of
decimal places as the sample standard deviation.
-To find the standard deviation of the diameter of wooden dowels, the manufacturer measures 19 randomly selected dowels and finds the standard deviation of the sample to be $s = 0.16$. Find the 95\% confidence interval for the population standard deviation $\sigma$.

Accepted Answer

A)  $0.13 < \sigma < 0.22$ 
B)  $0.11 < \sigma < 0.25$ 
C)  $0.15 < \sigma < 0.21$ 
D)  $0.12 < \sigma < 0.24$ 
A)  $0.13 < \sigma < 0.22$ 
B)  $0.11 < \sigma < 0.25$ 
C)  $0.15 < \sigma < 0.21$ 
D)  $0.12 < \sigma < 0.24$

Question 7

Express the confidence interval using the indicated format.
-Express the confidence interval $( 0.326,0.458 )$ in the form of $\hat { p } \pm \mathrm { E }$.

Accepted Answer

A)  $0.392 \pm 0.066$ 
B)  $0.326 \pm 0.132$ 
C)  $0.392 \pm 0.132$ 
D)  $0.326 \pm 0.066$ 
A)  $0.392 \pm 0.066$ 
B)  $0.326 \pm 0.132$ 
C)  $0.392 \pm 0.132$ 
D)  $0.326 \pm 0.066$

Question 8

Find the critical value $x _ { L } ^ { 2 }$ corresponding to a sample size of 4 and a confidence level of 95 percent.

Accepted Answer

A)  7.815 
B)  9.348 
C)  0.216 
D)  0.352 
A)  7.815 
B)  9.348 
C)  0.216 
D)  0.352

Question 9

To be able to say with 95% confidence level that the standard deviation of a data set is within 10% of the population's standard deviation, the number of observations within the data set must be greater than or equal to what quantity?

Accepted Answer

A)  805 
B)  335 
C)  250 
D)  192 
A)  805 
B)  335 
C)  250 
D)  192

Question 10

Find the critical value $\chi { } _ { \mathrm { R } } ^ { 2 }$ corresponding to a sample size of 19 and a confidence level of 99 percent.

Accepted Answer

A)  $34.805$ 
B)  $37.156$ 
C)  $6.265$ 
D)  $7.015$ 
A)  $34.805$ 
B)  $37.156$ 
C)  $6.265$ 
D)  $7.015$

Question 11

Use the given data to find the minimum sample size required to estimate the population proportion.
-Margin of error: 0.044; confidence level: 95%; $\hat{p}$ and $\hat{q}$ unknown

Accepted Answer

A)  635 
B)  497 
C)  352 
D)  405 
A)  635 
B)  497 
C)  352 
D)  405

Question 12

Do one of the following, as appropriate: (a) Find the critical value z?/2, (b) find the critical value t?/2, (c) state that neither the normal nor the t distribution applies.
-$91 \% ; \mathrm { n } = 45 ; \sigma$ is known; population appears to be very skewed.

Accepted Answer

A)  $\mathrm { z } _ { \alpha / 2 } = 1.75$ 
B)  $\mathrm { t } _ { \alpha / 2 } = 1.34$ 
C)  $\mathrm { z } _ { \alpha / 2 } = 1.70$ 
D)  $t _ { \alpha / 2 } = 1.645$ 
A)  $\mathrm { z } _ { \alpha / 2 } = 1.75$ 
B)  $\mathrm { t } _ { \alpha / 2 } = 1.34$ 
C)  $\mathrm { z } _ { \alpha / 2 } = 1.70$ 
D)  $t _ { \alpha / 2 } = 1.645$

Question 13

A one-sided confidence interval for $\mathrm { p }$ can be written as $\mathrm { p }  \hat { \mathrm { p } } - \mathrm { E }$ where the margin of error $\mathrm { E }$ is modified by replacing $\mathrm { z } \alpha / 2$ with $\mathrm { z } _ { \alpha }$. If a teacher wants to report that the fail rate on a test is at most $x$ with $90 \%$ confidence, construct the appropriate one-sided confidence interval. Assume that a simple random sample of 52 students results in 7 who fail the test.

Accepted Answer

A)  $p  0.074$ 
A)  $p  0.074$

Question 14

Why would manufacturers and businesses be interested in constructing a confidence interval for the population variance? Would manufacturers and businesses want large or small variances?

Accepted Answer

Manufacturers and businesses would be in

Question 15

Express the confidence interval using the indicated format.
-Express the confidence interval $0.62 < p < 0.72$ in the form of $\hat { p } \pm E$.

Accepted Answer

A)  $0.67 \pm 0.05$ 
B)  $0.62 \pm 0.1$ 
C)  $0.62 \pm 0.05$ 
D)  $0.67 \pm 0.1$ 
A)  $0.67 \pm 0.05$ 
B)  $0.62 \pm 0.1$ 
C)  $0.62 \pm 0.05$ 
D)  $0.67 \pm 0.1$

Question 16

Use the confidence level and sample data to find a confidence interval for estimating the population µ. Round your answer to the same number of decimal places as the sample mean.
-A laboratory tested 73 chicken eggs and found that the mean amount of cholesterol was 230 milligrams with $\sigma = 17.4$ milligrams. Construct a $95 \%$ confidence interval for the true mean cholesterol content, $\mu$, of all such eggs.

Accepted Answer

A)  $225 \mathrm { mg } < \mu < 234 \mathrm { mg }$ 
B)  $225 \mathrm { mg } < \mu < 233 \mathrm { mg }$ 
C)  $226 \mathrm { mg } < \mu < 234 \mathrm { mg }$ 
D)  $227 \mathrm { mg } < \mu < 235 \mathrm { mg }$ 
A)  $225 \mathrm { mg } < \mu < 234 \mathrm { mg }$ 
B)  $225 \mathrm { mg } < \mu < 233 \mathrm { mg }$ 
C)  $226 \mathrm { mg } < \mu < 234 \mathrm { mg }$ 
D)  $227 \mathrm { mg } < \mu < 235 \mathrm { mg }$

Question 17

$\mathrm { n } = 51 , \mathrm { x } = 23 ; 95 \%$ confidence

Accepted Answer

A)  $0.336 < p < 0.566$ 
B)  $0.313 < p < 0.589$ 
C)  $0.335 < p < 0.567$ 
D)  $0.314 < p < 0.588$ 
A)  $0.336 < p < 0.566$ 
B)  $0.313 < p < 0.589$ 
C)  $0.335 < p < 0.567$ 
D)  $0.314 < p < 0.588$

Question 18

Assume that a sample is used to estimate a population proportion p. Find the margin of error E that corresponds to the given statistics and confidence level. Round the margin of error to four decimal places.
-98% confidence; the sample size is 810, of which 20% are successes

Accepted Answer

A)  0.0275 
B)  0.0288 
C)  0.0327 
D)  0.0362 
A)  0.0275 
B)  0.0288 
C)  0.0327 
D)  0.0362

Question 19

Draw a diagram of the chi-square distribution. Discuss its shape and values.

Accepted Answer

The following graph illustrates examples

Question 20

A paper published the results of a poll. It stated that, based on a sample of 1000 married men, 51% of married men say that they would marry the same woman again. The margin of error was given as ±3 percentage points and the confidence level was given as 95%.
What does it mean that the margin of error was ±3 percentage points?

Accepted Answer

If 51% is used as an estimate of the per

Under what three conditions is it appropriate to use the t distribution in place of the standard normal distribution?

Describe the process for finding the confidence interval for a population proportion.

Express the confidence interval using the indicated format. -Express the confidence interval $- 0.061 < \mathrm { p } < 0.579$ in the form of $\hat { \mathrm { p } } \pm \mathrm { E }$ .

The following confidence interval is obtained for a population proportion, $p : ( 0.734,0.778 )$ Use these confidence interval limits to find the margin of error, E.

Express the confidence interval using the indicated format. -Express the confidence interval $( 0.326,0.458 )$ in the form of $\hat { p } \pm \mathrm { E }$ .

Find the critical value $x _ { L } ^ { 2 }$ corresponding to a sample size of 4 and a confidence level of 95 percent.

To be able to say with 95% confidence level that the standard deviation of a data set is within 10% of the population's standard deviation, the number of observations within the data set must be greater than or equal to what quantity?

Find the critical value $\chi { } _ { \mathrm { R } } ^ { 2 }$ corresponding to a sample size of 19 and a confidence level of 99 percent.

Use the given data to find the minimum sample size required to estimate the population proportion. -Margin of error: 0.044; confidence level: 95%; $\hat{p}$ and $\hat{q}$ unknown

Do one of the following, as appropriate: (a) Find the critical value z?/2, (b) find the critical value t?/2, (c) state that neither the normal nor the t distribution applies. - $91 \% ; \mathrm { n } = 45 ; \sigma$ is known; population appears to be very skewed.

Why would manufacturers and businesses be interested in constructing a confidence interval for the population variance? Would manufacturers and businesses want large or small variances?

Express the confidence interval using the indicated format. -Express the confidence interval $0.62 < p < 0.72$ in the form of $\hat { p } \pm E$ .

$\mathrm { n } = 51 , \mathrm { x } = 23 ; 95 \%$ confidence

Assume that a sample is used to estimate a population proportion p. Find the margin of error E that corresponds to the given statistics and confidence level. Round the margin of error to four decimal places. -98% confidence; the sample size is 810, of which 20% are successes

Draw a diagram of the chi-square distribution. Discuss its shape and values.

Introduction to Statistics

Summarizing and Graphing Data

Statistics for Describing, Exploring, and Comparing Data

Probability

Discrete Probability Distributions

Normal Probability Distributions

Hypothesis Testing

Inferences From Two Samples

Correlation and Regression

Chi-Square and Analysis of Variance

Filters

Exam 7: Estimates and Sample Size

Under what three conditions is it appropriate to use the t distribution in place of the standard normal distribution?

Describe the process for finding the confidence interval for a population proportion.

Express the confidence interval using the indicated format. -Express the confidence interval −0.061<p<0.579- 0.061 < \mathrm { p } < 0.579−0.061<p<0.579 in the form of p^±E\hat { \mathrm { p } } \pm \mathrm { E }p^​±E .

The following confidence interval is obtained for a population proportion, p:(0.734,0.778)p : ( 0.734,0.778 )p:(0.734,0.778) Use these confidence interval limits to find the margin of error, E.

Express the confidence interval using the indicated format. -Express the confidence interval (0.326,0.458)( 0.326,0.458 )(0.326,0.458) in the form of p^±E\hat { p } \pm \mathrm { E }p^​±E .

Find the critical value xL2x _ { L } ^ { 2 }xL2​ corresponding to a sample size of 4 and a confidence level of 95 percent.

To be able to say with 95% confidence level that the standard deviation of a data set is within 10% of the population's standard deviation, the number of observations within the data set must be greater than or equal to what quantity?

Find the critical value χR2\chi { } _ { \mathrm { R } } ^ { 2 }χR2​ corresponding to a sample size of 19 and a confidence level of 99 percent.

Use the given data to find the minimum sample size required to estimate the population proportion. -Margin of error: 0.044; confidence level: 95%; p^\hat{p}p^​ and q^\hat{q}q^​ unknown

Do one of the following, as appropriate: (a) Find the critical value z?/2, (b) find the critical value t?/2, (c) state that neither the normal nor the t distribution applies. - 91%;n=45;σ91 \% ; \mathrm { n } = 45 ; \sigma91%;n=45;σ is known; population appears to be very skewed.

Why would manufacturers and businesses be interested in constructing a confidence interval for the population variance? Would manufacturers and businesses want large or small variances?

Express the confidence interval using the indicated format. -Express the confidence interval 0.62<p<0.720.62 < p < 0.720.62<p<0.72 in the form of p^±E\hat { p } \pm Ep^​±E .

n=51,x=23;95%\mathrm { n } = 51 , \mathrm { x } = 23 ; 95 \%n=51,x=23;95% confidence

Assume that a sample is used to estimate a population proportion p. Find the margin of error E that corresponds to the given statistics and confidence level. Round the margin of error to four decimal places. -98% confidence; the sample size is 810, of which 20% are successes

Draw a diagram of the chi-square distribution. Discuss its shape and values.

Introduction to Statistics

Summarizing and Graphing Data

Statistics for Describing, Exploring, and Comparing Data

Probability

Discrete Probability Distributions

Normal Probability Distributions

Hypothesis Testing

Inferences From Two Samples

Correlation and Regression

Chi-Square and Analysis of Variance

Filters

Express the confidence interval using the indicated format. -Express the confidence interval $- 0.061 < \mathrm { p } < 0.579$ in the form of $\hat { \mathrm { p } } \pm \mathrm { E }$ .

The following confidence interval is obtained for a population proportion, $p : ( 0.734,0.778 )$ Use these confidence interval limits to find the margin of error, E.

Express the confidence interval using the indicated format. -Express the confidence interval $( 0.326,0.458 )$ in the form of $\hat { p } \pm \mathrm { E }$ .

Find the critical value $x _ { L } ^ { 2 }$ corresponding to a sample size of 4 and a confidence level of 95 percent.

Find the critical value $\chi { } _ { \mathrm { R } } ^ { 2 }$ corresponding to a sample size of 19 and a confidence level of 99 percent.

Use the given data to find the minimum sample size required to estimate the population proportion. -Margin of error: 0.044; confidence level: 95%; $\hat{p}$ and $\hat{q}$ unknown

Do one of the following, as appropriate: (a) Find the critical value z?/2, (b) find the critical value t?/2, (c) state that neither the normal nor the t distribution applies. - $91 \% ; \mathrm { n } = 45 ; \sigma$ is known; population appears to be very skewed.

Express the confidence interval using the indicated format. -Express the confidence interval $0.62 < p < 0.72$ in the form of $\hat { p } \pm E$ .

$\mathrm { n } = 51 , \mathrm { x } = 23 ; 95 \%$ confidence