Q: Suppose the true proportion of 'Remain' votes is $p = 0.5$. For a sample of $n = 546$, what is the standard deviation of the sampling distribution of $\hat{p}$?

$SD(\hat{p}) = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.5 \times 0.5}{546}} \approx 0.0214$

Q: A 95% confidence interval for a sample proportion $\hat{p}$ is constructed as:

$\hat{p} - 1.96 \times SE(\hat{p}) < p < \hat{p} + 1.96 \times SE(\hat{p})$

Q: Why do we use the $t$-distribution instead of the normal distribution when constructing confidence intervals for the mean of a continuous variable from a small sample?

Because the sample standard deviation $s$ is used to estimate the population standard deviation $\sigma$, introducing extra uncertainty.

Q: A casino promotion involves 50 players, with expected loss per player $\mu = 120$ and sample standard deviation $s = 100$. What is the standard error of the sample mean loss?

$SE(\bar{y}) = \frac{100}{\sqrt{50}} \approx 14.14$

Q: For a sample of $n = 13$ drivers, the average maximum speed is $\bar{y} = 50$ mph and $s = 11$ mph. What is the 95% confidence interval for the mean maximum speed? (Use $t^* \approx 2.160$ for $df = 12$)

$50 \pm 2.160 \times \frac{11}{\sqrt{13}} \approx 50 \pm 6.59$

Q: Which of the following is TRUE about the interpretation of a 95% confidence interval for a population mean?

About 95% of samples will produce intervals that contain the true mean, but we do not know if our specific interval does.

Question 1

Which of the following statements best describes the Central Limit Theorem (CLT) for sample means?

Accepted Answer

The CLT states that the distribution of sample means will be approximately normal if the sample size is large, regardless of the population's distribution.

Question 2

A poll of 546 UK voters found that 283 wanted to remain in the EU. What is the sample proportion $\hat{p}$ of 'Remain' votes?

Accepted Answer

$\hat{p} = \frac{283}{546} = 0.5183$

Question 3

Suppose the true proportion of 'Remain' votes is $p = 0.5$. For a sample of $n = 546$, what is the standard deviation of the sampling distribution of $\hat{p}$?

Accepted Answer

$SD(\hat{p}) = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.5 	imes 0.5}{546}} \approx 0.0214$

Question 4

If the sample proportion $\hat{p} = 0.518$ and the true proportion $p = 0.5$ with $SD(\hat{p}) = 0.0214$, what is the $z$-statistic for this sample?

Accepted Answer

$z = \frac{0.518 - 0.5}{0.0214} \approx 0.841$

Question 5

Which formula is used to estimate the standard error of the sample proportion when the true population proportion $p$ is unknown?

Accepted Answer

$SE(\hat{p}) = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

Question 6

A 95% confidence interval for a sample proportion $\hat{p}$ is constructed as:

Accepted Answer

$\hat{p} - 1.96 	imes SE(\hat{p}) < p < \hat{p} + 1.96 	imes SE(\hat{p})$

Question 7

Why do we use the $t$-distribution instead of the normal distribution when constructing confidence intervals for the mean of a continuous variable from a small sample?

Accepted Answer

Because the sample standard deviation $s$ is used to estimate the population standard deviation $\sigma$, introducing extra uncertainty.

Question 8

A casino promotion involves 50 players, with expected loss per player $\mu = 120$ and sample standard deviation $s = 100$. What is the standard error of the sample mean loss?

Accepted Answer

$SE(\bar{y}) = \frac{100}{\sqrt{50}} \approx 14.14$

Question 9

For a sample of $n = 13$ drivers, the average maximum speed is $\bar{y} = 50$ mph and $s = 11$ mph. What is the 95% confidence interval for the mean maximum speed? (Use $t^* \approx 2.160$ for $df = 12$)

Accepted Answer

$50 \pm 2.160 	imes \frac{11}{\sqrt{13}} \approx 50 \pm 6.59$

Question 10

Which of the following is TRUE about the interpretation of a 95% confidence interval for a population mean?

Accepted Answer

About 95% of samples will produce intervals that contain the true mean, but we do not know if our specific interval does.

Question 11

In the context of sampling distributions, what does 'sampling variation' refer to?

Accepted Answer

The fact that different random samples from the same population will produce different sample statistics.

Question 12

If the sample size $n$ increases, what happens to the standard error of the sample mean $SE(\bar{y})$?

Accepted Answer

It decreases, because $SE(\bar{y}) = \frac{s}{\sqrt{n}}$

Question 13

A mayor wants to estimate the average lease cost for businesses. From a sample of $n = 223$, the sample mean is $\bar{y} = 478,603$ and $s = 535,343$. What is the standard error of the mean?

Accepted Answer

$SE(\bar{y}) = \frac{535,343}{\sqrt{223}} \approx 35,849$

Question 14

Which condition is necessary for the Central Limit Theorem to apply to sample means?

Accepted Answer

The sample must be random and independent, and the sample size should be sufficiently large.

Question 15

For a binomial random variable $X$ with $E(X) = np$ and $SD(X) = \sqrt{np(1-p)}$, what is the expected value and variance of the sample proportion $\hat{p} = \frac{X}{n}$?

Accepted Answer

$E(\hat{p}) = p$, $Var(\hat{p}) = \frac{p(1-p)}{n}$

Question 16

Why do confidence intervals for means use the $t$-distribution for small samples?

Accepted Answer

Because the $t$-distribution accounts for the extra uncertainty from estimating the population standard deviation with the sample standard deviation.

Question 17

Which of the following is NOT a correct interpretation of a 95% confidence interval for a proportion?

Accepted Answer

There is a 95% probability that the true proportion is inside the interval.

Sampling Distributions, Central Limit Theorem, and Confidence Intervals: Study Notes for Statistics for Business

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