Question 1

When the hatching of young geese is very near, the father guards the nest to defend it from predators that may be attracted by the hatchlings noisy entrance into the world. The following data are the typical distances from the nest for 24 soon to be father geese. A biologist would like to construct a $95 \%$ confidence interval for the mean distance of future father geese from the nest during this period. (Distances are in $\mathrm { km }$.)
$\begin{array}{|l|l|l|l|l|l|}
\hline 2.0 & 4.4 & 4.5 & 4.5 & 4.7 & 4.9 \
\hline 4.9 & 5.0 & 5.2 & 5.3 & 5.3 & 5.4 \
\hline 5.4 & 5.4 & 5.5 & 5.5 & 5.6 & 5.6 \
\hline 5.6 & 5.6 & 5.7 & 5.8 & 5.9 & 6.3 \
\hline
\end{array}$
a) Using a graphical display of your choice, display the data in a way that will allow you to determine whether it would be appropriate to use a $95 \% t$ confidence interval to estimate the population mean.

b) Using the graphical display from part (a), would you advise that constructing a $95 \%$ confidence interval is appropriate? Explain the basis of your recommendation.

Accepted Answer

a)

b)
It would not be appropriate to construct a confidence interval, as there is an outlier and the data gives a clear indication of skew.

Question 2

A A random sample of size $n = 25$ is selected from a population with mean $\mu = 50$ and standard deviation $\sigma = 10$. Determine the mean and standard deviation of the sampling distribution of $\bar { x }$.

Accepted Answer

$$\begin{array} { l } 
\mu _ { \bar { x } } = \mu = 50 \
\sigma _ { \bar { x } } = \frac { \sigma } { \sqrt { n } } = \frac { 10 } { \sqrt { 25 } } = 2.0
\end{array}$$

Question 3

$$\mu _ { \bar { x } } \text { and } \sigma _ { \bar { x } } \text { are statistics. }$$

Accepted Answer

$\mu _ { \bar { x } }$ and $\sigma _ { \bar { x } }$ are parameters, specifically the mean and standard deviation of the sampling distribution of the sample mean, not statistics.

Question 4

$$\text { The standard deviation of the distribution of } \bar { x } \text { decreases as } n \text { increases. }$$

Accepted Answer

This is a known fact in statistics. The standard deviation of the sample mean decreases as sample size increases. This is due to the fact that as sample size increases, the sample mean becomes a more accurate estimate of the population mean, resulting in less variability between sample means.

Question 5

A simple random sample of 10 houses in a particular town was selected and the level of radon gas measured for each house in the sample. The sample mean was $7.7$ parts per million, and the standard deviation $2.98$ parts per million. Experience has shown that radon gas level is approximately normally distributed in similar populations.
a) Construct and interpret a $99 \%$ confidence interval for $\mu$.
b) The city standards for radon call for a radon level of no more than $6 \mathrm { ppm }$. Based on your answer to part (a), does it appear that the mean radon level for city houses is greater than 6 ppm? Provide statistical justification for your response.

Accepted Answer

The answer of A simple random sample of 10 houses...

Question 6

The level of significance of a test is the probability of making a type I
error.

Accepted Answer

The level of significance (alpha) is the probability of rejecting a true null hypothesis, which is equivalent to making a type I error.

Question 7

$$\bar { x } \text { is a statistic. }$$

Accepted Answer

$\bar{x}$ is a statistic because it is a numerical value that describes a characteristic of a sample.

Question 8

C Consider sampling from a normal population with a population mean $\mu$ and standard deviation $\sigma$. As the sample size, $n$, increases, some characteristics of the sampling distribution of the sample mean, $\bar { x }$, may change. Which of the following characteristics will change as $n$ increases, and what is the nature of the change?
a) The mean of the sampling distribution of $\bar { x }$

b) The standard deviation of the sampling distribution of $\bar { x }$

c) The shape of the sampling distribution of $\bar { x }$

Accepted Answer

The answer of C Consider sampling from a normal population...

Question 9

$$\text { The standard error of } \bar { x } \text { is } s \text {. }$$

Accepted Answer

The standard error of the mean, or SEM, is the standard deviation of the sampling distribution of the mean. It is calculated by dividing the standard deviation of the population by the square root of the sample size, not by the sample standard deviation (s) alone. Therefore, the statement is false.

Question 10

The $t$ confidence interval formula for estimating $\mu$ is appropriate when the population being sampled is approximately normally distributed.

Accepted Answer

The $t$ confidence interval formula assumes that the population being sampled is approximately normally distributed. If the population is not approximately normally distributed or if the sample size is small, alternative methods such as the Wilcoxon rank-sum test or the bootstrap method may be more appropriate.

Question 11

$$\text { A } t \text { curve is bell-shaped like the } z \text { curve but is less spread out. }$$

Accepted Answer

The t-distribution curve is bell-shaped like the z-distribution curve but is more spread out, especially with smaller sample sizes.

Question 12

C Consider sampling from a normal population with a population mean $\mu$ and standard deviation $\sigma$. As the sample size, $n$, increases, some characteristics of the sampling distribution of the sample mean, $\bar { x }$, may change. Which of the following characteristics will change as $n$ increases, and what is the nature of the change? a) The mean of the sampling distribution of $\bar { x }$

b) The standard deviation of the sampling distribution of $\bar { x }$

c) The shape of the sampling distribution of $\bar { x }$

Accepted Answer

The answer of C Consider sampling from a normal population...

Question 13

A A random sample of size $n = 36$ is selected from a population with mean $\mu = 50$ and standard deviation $\sigma = 12$. Determine the mean and standard deviation of the sampling distribution of $\bar { x }$.

Accepted Answer

The answer of A A random sample of size \(n...

Question 14

A statistic is a characteristic of a population.

Accepted Answer

A statistic is a characteristic of a sample, not a population. A characteristic of a population is called a parameter.

Question 15

All other things being equal, choosing a smaller value of α will increase
the probability of making a type II error.

Accepted Answer

When choosing a smaller value of α, we are increasing the threshold for rejecting the null hypothesis, which means we are becoming more conservative in our decision-making. This can lead to a higher likelihood of accepting a false null hypothesis, which in turn increases the probability of making a type II error.

Question 16

Local health authorities are concerned that the hectic pace in their city has elevated
the stress, and thus the blood pressure, of women. The authorities will study this
issue by measuring the blood pressure of a random sample of women in the city and
estimating the mean blood pressure, μ , of women in the city. Blood pressure is
approximately normally distributed in humans, and a previous study suggests the
population standard deviation of women's blood pressure is about 8 mm Hg.
Suppose the health authorities accept this value as a reasonable estimate of the
standard deviation of blood pressure of today's women in this city. If it is desired to
estimate μ to within 1.0 mm Hg with 95% confidence, what is the minimum sample
size that is necessary?

Accepted Answer

The answer of Local health authorities are concerned that the...

Question 17

A new species of sea crab has been discovered, and a study was conducted to determine whether or not the animal can regulate its temperature. If the animal can maintain a body temperature different from the surroundings, this would be considered evidence of regulating capability. Ten of these sea crabs were exposed to ambient temperatures of 24 degrees Celsius and their body temperatures were measured. The sample mean was $24.694$ degrees, and the standard deviation $0.277$ degrees.

For purposes of this example, assume that it is reasonable to regard these 10 crabs as a random sample from the population of all crabs of this species. A plot of the data indicated that the assumption of approximately normal body temperatures was credible.
a) Construct and interpret a $99 \%$ confidence interval for the mean body temperature, $\mu$.

b) Based on your answer to part (a), does it appear that the crabs are able to regulate their body temperature? Provide statistical justification for your response.

Accepted Answer

The answer of A new species of sea crab has...

Question 18

If the null hypothesis is not rejected, there is strong statistical evidence
that the null hypothesis is true.

Accepted Answer

If the null hypothesis is not rejected, it only means that there is not enough evidence to reject it. It does not necessarily mean that the null hypothesis is true. There may be other factors or limitations in the study that prevent the rejection of the null hypothesis.

Question 19

We say that the result of a hypothesis test is "statistically significant"
when the P-value is less than or equal toα.

Accepted Answer

This is the correct definition of statistical significance.

Question 20

As $n$ increases, the mean of the sampling distribution of $\bar { x }$ gets closer to $\mu$.

Accepted Answer

The answer of As $n$ increases, the mean of the...

Quiz 12: Asking and Answering Questions About a Population Mean

A A random sample of size $n = 25$ is selected from a population with mean $\mu = 50$ and standard deviation $\sigma = 10$ . Determine the mean and standard deviation of the sampling distribution of $\bar { x }$ .

$\mu _ { \bar { x } } \text { and } \sigma _ { \bar { x } } \text { are statistics. }$

$\text { The standard deviation of the distribution of } \bar { x } \text { decreases as } n \text { increases. }$

The level of significance of a test is the probability of making a type I error.

$\bar { x } \text { is a statistic. }$

$\text { The standard error of } \bar { x } \text { is } s \text {. }$

The $t$ confidence interval formula for estimating $\mu$ is appropriate when the population being sampled is approximately normally distributed.

$\text { A } t \text { curve is bell-shaped like the } z \text { curve but is less spread out. }$

A A random sample of size $n = 36$ is selected from a population with mean $\mu = 50$ and standard deviation $\sigma = 12$ . Determine the mean and standard deviation of the sampling distribution of $\bar { x }$ .

A statistic is a characteristic of a population.

All other things being equal, choosing a smaller value of α will increase the probability of making a type II error.

If the null hypothesis is not rejected, there is strong statistical evidence that the null hypothesis is true.

We say that the result of a hypothesis test is "statistically significant" when the P-value is less than or equal toα.

As $n$ increases, the mean of the sampling distribution of $\bar { x }$ gets closer to $\mu$ .

Quiz 12: Asking and Answering Questions About a Population Mean

A A random sample of size n=25n = 25n=25 is selected from a population with mean μ=50\mu = 50μ=50 and standard deviation σ=10\sigma = 10σ=10 . Determine the mean and standard deviation of the sampling distribution of xˉ\bar { x }xˉ .

μxˉ and σxˉ are statistics. \mu _ { \bar { x } } \text { and } \sigma _ { \bar { x } } \text { are statistics. }μxˉ​ and σxˉ​ are statistics.

The standard deviation of the distribution of xˉ decreases as n increases. \text { The standard deviation of the distribution of } \bar { x } \text { decreases as } n \text { increases. } The standard deviation of the distribution of xˉ decreases as n increases.

The level of significance of a test is the probability of making a type I error.

xˉ is a statistic. \bar { x } \text { is a statistic. }xˉ is a statistic.

The standard error of xˉ is s. \text { The standard error of } \bar { x } \text { is } s \text {. } The standard error of xˉ is s.

The ttt confidence interval formula for estimating μ\muμ is appropriate when the population being sampled is approximately normally distributed.

A t curve is bell-shaped like the z curve but is less spread out. \text { A } t \text { curve is bell-shaped like the } z \text { curve but is less spread out. } A t curve is bell-shaped like the z curve but is less spread out.

A A random sample of size n=36n = 36n=36 is selected from a population with mean μ=50\mu = 50μ=50 and standard deviation σ=12\sigma = 12σ=12 . Determine the mean and standard deviation of the sampling distribution of xˉ\bar { x }xˉ .

A statistic is a characteristic of a population.

All other things being equal, choosing a smaller value of α will increase the probability of making a type II error.

If the null hypothesis is not rejected, there is strong statistical evidence that the null hypothesis is true.

We say that the result of a hypothesis test is "statistically significant" when the P-value is less than or equal toα.

As nnn increases, the mean of the sampling distribution of xˉ\bar { x }xˉ gets closer to μ\muμ .

A A random sample of size $n = 25$ is selected from a population with mean $\mu = 50$ and standard deviation $\sigma = 10$ . Determine the mean and standard deviation of the sampling distribution of $\bar { x }$ .

$\mu _ { \bar { x } } \text { and } \sigma _ { \bar { x } } \text { are statistics. }$

$\text { The standard deviation of the distribution of } \bar { x } \text { decreases as } n \text { increases. }$

$\bar { x } \text { is a statistic. }$

$\text { The standard error of } \bar { x } \text { is } s \text {. }$

The $t$ confidence interval formula for estimating $\mu$ is appropriate when the population being sampled is approximately normally distributed.

$\text { A } t \text { curve is bell-shaped like the } z \text { curve but is less spread out. }$

A A random sample of size $n = 36$ is selected from a population with mean $\mu = 50$ and standard deviation $\sigma = 12$ . Determine the mean and standard deviation of the sampling distribution of $\bar { x }$ .

As $n$ increases, the mean of the sampling distribution of $\bar { x }$ gets closer to $\mu$ .