Question 1

50 people are selected randomly from a certain population and it is found that 18 people in the sample are over 6 feet tall. What is the point estimate of the proportion of people in the population who are over 6 feet tall?

Accepted Answer

The point estimate of the proportion is calculated by dividing the number of people over 6 feet tall by the total number of people in the sample: 18/50 = 0.36.

Question 2

A $99 \%$ confidence interval (in inches) for the mean height of a population is $66.2 < \mu < 67.6$. This result is based on a sample of size 144 . Construct the $95 \%$ confidence interval. (Hint: you will first need to find the sample mean and sample standard deviation).

Accepted Answer

To construct a 95% confidence interval, we first find the sample mean, which is the midpoint of the 99% confidence interval: (66.2 + 67.6) / 2 = 66.9. The margin of error for the 99% confidence interval is (67.6 - 66.2) / 2 = 0.7. Using the z-scores for 99% (2.576) and 95% (1.96) confidence levels, we find the standard deviation: 0.7 = 2.576 * (σ/√144). Solving for σ gives σ = 0.7 * √144 / 2.576. For the 95% confidence interval, the margin of error is 1.96 * (σ/√144). Calculating this gives the interval 66.3 < μ < 67.5.

Question 3

Use the given degree of confidence and sample data to construct a confidence interval for the population mean µ. Assume that the population has a normal distribution.
-A sociologist develops a test to measure attitudes towards public transportation, and 27 randomly selected subjects are given the test. Their mean score is 76.2 and their standard deviation is 21.4.
Construct the 95% confidence interval for the mean score of all such subjects.

Accepted Answer

The answer of Use the given degree of confidence and...

Question 4

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.
-Test scores: $\mathrm { n } = 79 , \overline { \mathrm { x } } = 65.0 , \sigma = 5.0 ; 98 \%$ confidence

Accepted Answer

The answer of Use the confidence level and sample data...

Question 5

The Bide-a-While efficiency hotel, which caters to business workers who stay for extended periods of time (weeks or months), offers room service. In a small study of 35 randomly selected room service orders, the 95% confidence interval for mean delivery time for room service is $24.8 < \mu < 29.6$ minutes. The marketing director is trying to determine if she can advertise "room service in under 30 minutes, or the order is free." How would you advise her?

Accepted Answer

The answer of The Bide-a-While efficiency hotel, which caters to...

Question 6

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 group of 64 randomly selected students have a mean score of $38.6$ with a standard deviation of $4.9$ on a placement test. What is the $90 \%$ confidence interval for the mean score, $\mu$, of all students taking the test?

Accepted Answer

The 90% confidence interval is calculated using the formula: $\bar{x} \pm z \times \frac{\sigma}{\sqrt{n}}$. Here, $\bar{x} = 38.6$, $\sigma = 4.9$, $n = 64$, and the z-value for 90% confidence is approximately 1.645. The interval is $38.6 \pm 1.645 \times \frac{4.9}{\sqrt{64}}$, which simplifies to $37.6 < \mu < 39.6$.

Question 7

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

Accepted Answer

The answer of Under what three conditions is it appropriate...

Question 8

Use the given data to find the minimum sample size required to estimate the population proportion.
-Margin of error: 0.018; confidence level: $99 \% ; \hat { \mathrm { p } } \text { and } \hat { \mathrm { q } } \text { unknown }$

Accepted Answer

The answer of Use the given data to find the...

Question 9

Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p.
-$\mathrm { n } = 110 , \mathrm { x } = 68 ; 88 \%$ confidence

Accepted Answer

The answer of Use the given degree of confidence and...

Question 10

In constructing a confidence interval for $\sigma$ or $\sigma ^ { 2 }$, a table is used to find the critical values $\chi _ { L } ^ { 2 }$ and $\chi \underset { R } { 2 }$ for values of $n \leq 101$. For larger values of $n , \chi _ { L } ^ { 2 }$ and $\chi { } _ { \mathrm { R } } ^ { 2 }$ can be approximated by using the following formula: $\chi ^ { 2 } = \frac { 1 } { 2 } [ \pm \mathrm { z } \alpha / 2 + \sqrt { 2 \mathrm { k } - 1 } ] ^ { 2 }$ where $\mathrm { k }$ is the number of degrees of freedom and $\mathrm { z } \alpha / 2$ is the critical z score. Construct the $90 \%$ confidence interval for $\sigma$ using the following sample data: a sample of size $\mathrm { n } = 234$ yields a mean weight of $155 \mathrm { lb }$ and a standard deviation of $26.0 \mathrm { lb }$. Round the confidence interval limits to the nearest hundredth.

Accepted Answer

The formula for the confidence interval for $\sigma$ uses the chi-squared distribution. For large $n$, the approximation formula is used. With $n = 234$, the degrees of freedom $k = 233$. Using the approximation formula and the critical z-score for a 90% confidence interval, the calculated interval is closest to option B.

Question 11

A radio show host asked people to call in and say whether they support new legislation to promote cleaner sources of energy. Based on this sample, she constructed a confidence interval to estimate the proportion of all listeners to her show who support the legislation.
Is the confidence interval likely to give a good estimate of the proportion of her listeners who support the legislation?

Accepted Answer

The answer of A radio show host asked people to...

Question 12

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

The answer of A paper published the results of a...

Question 13

Use the given data to find the minimum sample size required to estimate the population proportion.
-Margin of error: 0.008; confidence level: 99%; from a prior study, $\hat{p}$ is estimated by 0.208.

Accepted Answer

The answer of Use the given data to find the...

Question 14

Define a point estimate. What is the best point estimate for $\mu$ ?

Accepted Answer

The answer of Define a point estimate. What is the...

Question 15

Mark wanted to estimate the mean number of years of education of adults in his city. He waited outside a public library and interviewed every tenth adult leaving. Based on this sample, he constructed a confidence interval for the mean number of years of education of adults in the city. Do you think this confidence interval will give a good estimate? Why or why not?

Accepted Answer

The answer of Mark wanted to estimate the mean number...

Question 16

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

Accepted Answer

The answer of Draw a diagram of the chi-square distribution....

Question 17

Define confidence interval and degree of confidence. Make up an example of a confidence interval and interpret the result.

Accepted Answer

The answer of Define confidence interval and degree of confidence....

Question 18

Of 366 randomly selected medical students, 27 said that they planned to work in a rural community. Find a 95% confidence interval for the true proportion of all medical students who plan to work in a rural community.

Accepted Answer

The sample proportion is 27/366. Using the formula for a confidence interval for a proportion, the interval is calculated as $ \hat{p} \pm Z \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} $, where $ \hat{p} = \frac{27}{366} $, $ Z $ is the Z-score for 95% confidence (1.96), and $ n = 366 $. This calculation results in the interval $0.0470 < p < 0.101$.

Question 19

Use the given data to find the minimum sample size required to estimate the population proportion.
-Margin of error: $0.028$; confidence level: $99 \% ; \hat { \mathrm { p } }$ and $\hat { \mathrm { q } }$ unknown

Accepted Answer

The answer of Use the given data to find the...

Question 20

Identify the correct distribution (z, t, or neither) for each of the following. $$\begin{array} { | l | l | l | l | } 
\hline \text { Sample Size } & \begin{array} { l } 
\text { Standard } \
\text { deviation }
\end{array} & \begin{array} { l } 
\text { Shape of the } \
\text { distribution }
\end{array} & \text { z or t or neither } \
\hline n = 35 & s = 4.5 & \text { Somewhat skewed } & \
\hline n = 20 & s = 4.5 & \text { Bell shaped } & \
\hline n = 25 & \sigma = 4.5 & \text { Bell shaped } & \
\hline n = 20 & \sigma = 4.5 & \text { Extremely skewed } & \
\hline
\end{array}$$

Accepted Answer

The answer of Identify the correct distribution (z, t, or...

Quiz 7: Estimates and Sample Size

50 people are selected randomly from a certain population and it is found that 18 people in the sample are over 6 feet tall. What is the point estimate of the proportion of people in the population who are over 6 feet tall?

A $99 \%$ confidence interval (in inches) for the mean height of a population is $66.2 < \mu < 67.6$ . This result is based on a sample of size 144 . Construct the $95 \%$ confidence interval. (Hint: you will first need to find the sample mean and sample standard deviation) .

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. -Test scores: $\mathrm { n } = 79 , \overline { \mathrm { x } } = 65.0 , \sigma = 5.0 ; 98 \%$ confidence

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

Use the given data to find the minimum sample size required to estimate the population proportion. -Margin of error: 0.018; confidence level: $99 \% ; \hat { \mathrm { p } } \text { and } \hat { \mathrm { q } } \text { unknown }$

Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p. - $\mathrm { n } = 110 , \mathrm { x } = 68 ; 88 \%$ confidence

Use the given data to find the minimum sample size required to estimate the population proportion. -Margin of error: 0.008; confidence level: 99%; from a prior study, $\hat{p}$ is estimated by 0.208.

Define a point estimate. What is the best point estimate for $\mu$ ?

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

Define confidence interval and degree of confidence. Make up an example of a confidence interval and interpret the result.

Of 366 randomly selected medical students, 27 said that they planned to work in a rural community. Find a 95% confidence interval for the true proportion of all medical students who plan to work in a rural community.

Use the given data to find the minimum sample size required to estimate the population proportion. -Margin of error: $0.028$ ; confidence level: $99 \% ; \hat { \mathrm { p } }$ and $\hat { \mathrm { q } }$ unknown

Quiz 7: Estimates and Sample Size

50 people are selected randomly from a certain population and it is found that 18 people in the sample are over 6 feet tall. What is the point estimate of the proportion of people in the population who are over 6 feet tall?

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

Use the given data to find the minimum sample size required to estimate the population proportion. -Margin of error: 0.018; confidence level: 99%;p^ and q^ unknown 99 \% ; \hat { \mathrm { p } } \text { and } \hat { \mathrm { q } } \text { unknown }99%;p^​ and q^​ unknown

Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p. - n=110,x=68;88%\mathrm { n } = 110 , \mathrm { x } = 68 ; 88 \%n=110,x=68;88% confidence

Use the given data to find the minimum sample size required to estimate the population proportion. -Margin of error: 0.008; confidence level: 99%; from a prior study, p^\hat{p}p^​ is estimated by 0.208.

Define a point estimate. What is the best point estimate for μ\muμ ?

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

Define confidence interval and degree of confidence. Make up an example of a confidence interval and interpret the result.

Of 366 randomly selected medical students, 27 said that they planned to work in a rural community. Find a 95% confidence interval for the true proportion of all medical students who plan to work in a rural community.

Use the given data to find the minimum sample size required to estimate the population proportion. -Margin of error: 0.0280.0280.028 ; confidence level: 99%;p^99 \% ; \hat { \mathrm { p } }99%;p^​ and q^\hat { \mathrm { q } }q^​ unknown

Use the given data to find the minimum sample size required to estimate the population proportion. -Margin of error: 0.018; confidence level: $99 \% ; \hat { \mathrm { p } } \text { and } \hat { \mathrm { q } } \text { unknown }$

Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p. - $\mathrm { n } = 110 , \mathrm { x } = 68 ; 88 \%$ confidence

Use the given data to find the minimum sample size required to estimate the population proportion. -Margin of error: 0.008; confidence level: 99%; from a prior study, $\hat{p}$ is estimated by 0.208.

Define a point estimate. What is the best point estimate for $\mu$ ?

Use the given data to find the minimum sample size required to estimate the population proportion. -Margin of error: $0.028$ ; confidence level: $99 \% ; \hat { \mathrm { p } }$ and $\hat { \mathrm { q } }$ unknown