AP Statistics - SD: Sampling Distributions

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AP Statistics - SD: Sampling Distributions 0/31
1 Probability > Sampling distributions · Level 3
The owner of a bagel shop, who is the father of an AP Statistics student, advertises that the price of a dozen bagels on any given day will be randomly picked using a normal distribution with a mean of \(10.00\) and a standard deviation of \(52\).
A
\(P\left(z > \dfrac{10.40-10.00}{0.50}\right)\)
B
\(P\left(z > \dfrac{52.00-50.00}{0.50}\right)\)
C
\(2P\left(z > \dfrac{52.00-50.00}{0.50}\right)\)
D
\(P\left(z > \dfrac{10.40-10.00}{\dfrac{0.50}{\sqrt{5}}}\right)\)
E
\(P\left(z > \dfrac{52.00-50.00}{\dfrac{0.50}{\sqrt{5}}}\right)\)
2 Probability > Sampling distributions · Level 3
6.7 percent of women graduate college with STEM degrees. In a simple random sample (SRS) of 1,000 women, what is the probability that more than 8 percent will graduate college with a STEM degree?
A
0.033
B
0.05
C
0.067
D
0.10
E
0.933
3 Probability > Sampling distributions · Level 3
Which of the following statements is incorrect?
A
The larger the sample is, the larger the spread is in the sampling distribution.
B
Provided that the population size is significantly greater than the sample size, the spread of a sampling distribution does not depend on the population size.
C
Bias has to do with the center, not the spread, of a sampling distribution.
D
Sample distribution and sampling distribution refer to different things.
E
The larger the sample is, the closer the sample distribution generally becomes to the population distribution.
4 Probability > Sampling distributions · Level 3
Suppose 15 percent of the mines in a particular Alaska region will strike gold. In a random sample of 200 mines in this region, what is the probability that more than 18 percent will strike gold?
A
\(P(z > \dfrac{0.18-0.15}{\sqrt{200(0.15)(0.85)}})\)
B
\(P(z > \dfrac{0.18-0.15}{\sqrt{200(0.5)(0.5)}})\)
C
\(P(z > \dfrac{0.18-0.15}{(0.15)\dfrac{0.85}{\sqrt{200}}})\)
D
\(P(z > \dfrac{0.18-0.15}{\sqrt{\dfrac{(0.15)(0.85)}{200}}})\)
E
\(P(z > \dfrac{0.18-0.15}{\sqrt{\dfrac{(0.5)(0.5)}{200}}})\)
5 Probability > Sampling distributions · Level 3
Which of the following statements is incorrect?
A
The sampling distribution of \(\overline{x}\) has a mean equal to the population mean \(\mu\) even if the sample size \(n\) is small.
B
The sampling distribution of \(\overline{x}\) has a standard deviation of \(\dfrac{\sigma}{\sqrt{n}}\) even if the population is not normally distributed and even if \(n\) is small.
C
The sampling distribution of \(\overline{x}\) is normal, no matter what \(n\) is, if the population has a normal distribution.
D
When \(n\) is large, the sampling distribution of \(\overline{x}\) is approximately normal even if the population is not normally distributed.
E
Even if the observations are not independent, the central limit theorem applies as long as \(n\) is large enough.
6 Probability > Sampling distributions · Level 3
It is known that 73 percent of the employees at one factory are women, while 61 percent of the employees of a second factory are women. In a simple random sample (SRS) of 115 employees from the first factory and an independent SRS of 80 employees from the second, what is the probability that the difference between the percentages of women picked (first factory minus second) is more than 10 percent?
A
\(P(z > \dfrac{0.10-0.12}{\sqrt{(0.5)(0.5)\left(\dfrac{1}{115}+\dfrac{1}{80}\right)}})\)
B
\(P(z > \dfrac{0.10-0.12}{\sqrt{(0.67)(0.33)\left(\dfrac{1}{115}+\dfrac{1}{80}\right)}})\)
C
\(P(z > \dfrac{0.10-0.12}{\sqrt{\dfrac{(0.73)(0.27)}{115}+\dfrac{(0.61)(0.39)}{80}}})\)
D
\(P(z > \dfrac{0.10-0.12}{\dfrac{(0.73)(0.27)}{\sqrt{115}}+\dfrac{(0.61)(0.39)}{\sqrt{80}}})\)
E
\(P(z > \dfrac{0.10-0.12}{\sqrt{\dfrac{84+49}{115+80}\left(1-\dfrac{84+49}{115+80}\right)\left(\dfrac{1}{115}+\dfrac{1}{80}\right)}})\)
7 Probability > Sampling distributions · Level 3
A population is normally distributed with a mean of 37. Consider all samples of size 8. The variable \(\dfrac{\overline{x} - 37}{\dfrac{s}{\sqrt{8}}}\)
A
has a normal distribution.
B
has a t-distribution with df = 8
C
has a t-distribution with df = 7
D
has neither a normal distribution nor a t-distribution
E
has either a normal distribution or a t-distribution depending on the characteristics of the population standard deviation
8 Probability > Sampling distributions · Level 3
Which of the following statements is incorrect?
A
Sample statistics are used to make inferences about population parameters.
B
Statistics from smaller samples have more variability than those from larger samples.
C
Parameters are fixed, while statistics vary depending on which sample is chosen.
D
As the sample size n becomes larger, the sample distribution becomes closer to a normal distribution.
E
All of the above are true statements.
9 Probability > Sampling distributions · Level 3
Which of the following is a true statement?
A
The sampling distribution of \(\hat{p}\) has a mean that can vary from the population proportion p by approximately 1.96 standard deviations.
B
The sampling distribution of \(\hat{p}\) has a standard deviation equal to \(\sqrt{n p(1 - p)}\).
C
The sampling distribution of \(\hat{p}\) is considered close to normal provided that n ≥ 30.
D
The sample proportion is a random variable with a probability distribution.
E
All of the above are true statements.
10 Probability > Sampling distributions · Level 3
Which of the following is a true statement?
A
The sampling distribution of the difference \(\overline{x}_1 - \overline{x}_2\) has a mean equal to the difference of the population means.
B
The sampling distribution of the difference \(\overline{x}_1 - \overline{x}_2\) has a standard deviation equal to the sum of the population standard deviations.
C
The sampling distribution of the difference \(\overline{x}_1 - \overline{x}_2\) has a standard deviation equal to the difference of the population standard deviations.
D
The sample sizes must be equal in two-sample inference.
E
As long as the sample sizes are large enough (for example, each at least 30), the samples in two-sample inference do not have to be independent.
11 Probability > Sampling distributions · Level 3
The calories in butterscotch hard candies are normally distributed with a mean of 23.1 and a standard deviation of 0.6. In a random sample of 10 butterscotch candies, what is the probability that the total calories are between 228 and 232?
A
\(P\left(\dfrac{228 - 231}{0.6} < z < \dfrac{232 - 231}{0.6}\right)\)
B
\(P\left(\dfrac{228 - 231}{6} < z < \dfrac{232 - 231}{6}\right)\)
C
\(P\left(\dfrac{22.8 - 23.1}{0.6} < z < \dfrac{23.2 - 23.1}{0.6}\right)\)
D
\(P\left(\dfrac{22.8 - 23.1}{\dfrac{0.6}{\sqrt{10}}} < z < \dfrac{23.2 - 23.1}{\dfrac{0.6}{\sqrt{10}}}\right)\)
E
\(P\left(\dfrac{228 - 231}{\dfrac{0.6}{\sqrt{10}}} < z < \dfrac{232 - 231}{\dfrac{0.6}{\sqrt{10}}}\right)\)
12 Probability > Sampling distributions · Level 3
Which of the following are unbiased estimators for the corresponding population parameters?
I. Sample means
II. Sample proportions
III. Difference of sample means
IV. Difference of sample proportions
A
None are unbiased.
B
I and II only
C
I and III only
D
III and IV only
E
All are unbiased.
13 Probability > Sampling distributions · Level 3
Which one of these histograms represents the sampling distribution of \(\hat{p}\) for p = 0.6 and n = 150?
A
Choice A
B
Choice B
C
Choice C
D
Choice D
E
Choice E
14 Probability > Sampling distributions · Level 3
Which of the following statements is true?
A
The mean of the set of sample means varies inversely as the square root of the size of the samples.
B
The variance of the set of sample means varies directly as the size of the samples and inversely as the variance of the original population.
C
The standard deviation of the set of sample means varies directly as the standard deviation of the original population and inversely as the square root of the size of the samples.
D
The larger the sample size, the larger the variance of the set of sample means.
E
One must double the sample size in order to cut the standard deviation of \(\overline{x}\) in half.
15 Probability > Sampling distributions · Level 3
Surveys have shown that 30 percent of people refuse to use public toilets. A random sample of 200 people is selected, and the proportion of those people who refuse to use public toilets is noted. This is repeated 100 times, and a dotplot is constructed of the 100 sample proportions. Which of the following best describes the standard deviation of the data in the dotplot?
A
\(\sqrt{100(0.3)(0.7)}\)
B
\(\sqrt{200(0.3)(0.7)}\)
C
\(\sqrt{300(0.3)(0.7)}\)
D
\(\sqrt{\dfrac{(0.3)(0.7)}{100}}\)
E
\(\sqrt{\dfrac{(0.3)(0.7)}{200}}\)
16 Probability > Sampling distributions · Level 3
Past surveys have shown that 9 percent of high school students and 12 percent of college students skip breakfast. A polling agency picks independent random samples of 100 high school students and of 100 college students. The agency surveys each group as to the proportion of students who skip breakfast. The difference in proportions is noted. This is repeated 400 times, and a dotplot is constructed of the 400 differences of proportions. Which of the following best describes the standard deviation of the data in the dotplot?
A
\(\sqrt{100(0.09)(0.91) + 100(0.12)(0.88)}\)
B
\(\sqrt{400(0.09)(0.91) + 400(0.12)(0.88)}\)
C
\(\sqrt{\dfrac{(0.09)(0.91)}{100} + \dfrac{(0.12)(0.88)}{100}}\)
D
\(\sqrt{\dfrac{(0.09)(0.91)}{400} + \dfrac{(0.12)(0.88)}{400}}\)
E
\(\sqrt{(0.105)(0.895)\left(\dfrac{1}{100} + \dfrac{1}{100}\right)}\)
17 Probability > Sampling distributions · Level 3
The distribution of weights of "18-ounce" boxes of corn flakes is given by the following histogram.
The distribution has a mean of 18.96 ounces with a standard deviation of 2.31 ounces. If 50 random samples of 12 boxes each are picked and if the mean weight of each sample is found, which of the following is most likely to represent the distribution of the sample means?
question image
A
Choice A
B
Choice B
C
Choice C
D
Choice D
E
None of the above gives a reasonable histogram for the sampling distribution.
18 Probability > Sampling distributions · Level 3
Which of the sample sizes \(n\) and population proportions \(p\) below would result in the greatest standard deviation for the sampling distribution of \(\hat{p}\)?
A
\(n = 50\) and \(p = 0.1\)
B
\(n = 50\) and \(p = 0.4\)
C
\(n = 250\) and \(p = 0.1\)
D
\(n = 250\) and \(p = 0.6\)
E
\(n = 250\) and \(p = 0.95\)
19 Probability > Sampling distributions · Level 3
The sampling distribution of the sample mean is close to the normal distribution
A
only if the parent population is unimodal, is not badly skewed, and does not have outliers
B
no matter the distribution of the parent population or what the value of \(n\)
C
if \(n\) is large, no matter the distribution of the parent population
D
if the standard deviation of the parent population is known
E
only if both \(n\) is large and the parent population has a normal distribution
20 Probability > Sampling distributions · Level 3
Suppose that 48 percent of high school students and 44 percent of college students would answer yes to the question "Do you enjoy Drake's music?" In a simulation, a random sample of 100 high school students and 100 college students is selected. The difference in sample proportions of those who answered yes, \(\hat{p}_{\text{high school}} - \hat{p}_{\text{college}}\), is calculated. The procedure is repeated 10,000 times. Which of the histograms below is most likely to result as the simulated sampling distribution of the difference in proportions?
A
Choice A
B
Choice B
C
Choice C
D
Choice D
E
Choice E
21 Probability > Sampling distributions · Level 3
The body temperatures of healthy elderly adults follow a skewed left distribution with a mean of 37°C and a standard deviation of 0.4°C. A random sample of 100 healthy adults is selected, and their mean body temperature is recorded. This is repeated 500 times, and a dotplot is constructed of the 500 sample means. Which of the following best describes the dotplot?
A
Skewed left with a mean of 37°C and a standard deviation of 0.0179°C
B
Skewed left with a mean of 37°C and a standard deviation of 0.04°C
C
Skewed left with a mean of 37°C and a standard deviation of 0.4°C
D
Approximately normal with a mean of 37°C and a standard deviation of 0.0179°C
E
Approximately normal with a mean of 37°C and a standard deviation of 0.04°C
22 Probability > Sampling distributions · Level 3
Which of the following is a biased estimator?
A
Sampling distribution of proportions
B
Sampling distribution of means
C
Sampling distribution of slopes
D
Sampling distribution of maxima
E
All of the above are unbiased.
23 Probability > Sampling distributions · Level 3
A simulation is conducted by using 25 fair dice whose faces are numbered 1 through 6, tossing them all at once, and averaging the 25 numbers showing face up. This is repeated 400 times. Which of these best describes the distribution being simulated?
A
A sampling distribution of a sample proportion with \(\mu_{\hat{p}} = \dfrac{1}{6}\) and \(\sigma_{\hat{p}} = \sqrt{\dfrac{\left(\dfrac{1}{6}\right)\left(\dfrac{5}{6}\right)}{25}}\)
B
A sampling distribution of a sample proportion with \(\mu_{\hat{p}} = \dfrac{1}{6}\) and \(\sigma_{\hat{p}} = \sqrt{\dfrac{\left(\dfrac{1}{6}\right)\left(\dfrac{5}{6}\right)}{400}}\)
C
A sampling distribution of a sample mean with \(\mu_{\overline{x}} = 3.5\) and \(\sigma_{\overline{x}} = 1.708\)
D
A sampling distribution of a sample mean with \(\mu_{\overline{x}} = 3.5\) and \(\sigma_{\overline{x}} = \dfrac{1.708}{\sqrt{25}}\)
E
A sampling distribution of a sample mean with \(\mu_{\overline{x}} = 3.5\) and \(\sigma_{\overline{x}} = \dfrac{1.708}{\sqrt{400}}\)
24 Probability > Sampling distributions · Level 3
Cycle times for express washing for Brand A dishwashers have a mean of 42.5 minutes with a standard deviation of 0.6 minutes, while times for the same cycle in Brand B dishwashers have a mean of 41.2 minutes with a standard deviation of 0.5 minutes. Brand A and Brand B dishwashers are each sent through an express washing 100 times. What is the probability that the mean difference (A minus B) is greater than 1 minute?
A
\(P\left(z > \dfrac{1-1.3}{\sqrt{0.6^2+0.5^2}}\right)\)
B
\(P\left(z > \dfrac{1-1.3}{\sqrt{\dfrac{0.6^2}{100}+\dfrac{0.5^2}{100}}}\right)\)
C
\(P\left(z > \dfrac{1-1.3}{\dfrac{0.6}{\sqrt{100}}+\dfrac{0.5}{\sqrt{100}}}\right)\)
D
\(P\left(z > \dfrac{1-1.3}{\sqrt{\dfrac{0.6^2+0.5^2}{100}}}\right)\) \(P\left(z > \dfrac{1-1.13}{\sqrt{\dfrac{0.62+0.52}{100+100}}}\right)\)
25 Probability > Sampling distributions · Level 3
The mean waist size for American men is 39.7 inches with a standard deviation of 3.8 inches. In a random sample of 200 men, what is the probability that the mean waist size is less than 35 inches?
A
\(P(z < 35)\)
B
\(P\left(z < \dfrac{35-39.7}{3.8}\right)\)
C
\(P\left(z < \dfrac{35-39.7}{\dfrac{3.8}{\sqrt{200}}}\right)\)
D
\(P\left(z < \dfrac{35-39.7}{\sqrt{\dfrac{38}{200}}}\right)\)
E
\(P\left(z < \dfrac{35-39.7}{\sqrt{\dfrac{3.8}{200}}}\right)\)
26 Probability > Sampling distributions · Level 3
Based on past studies, a credit card company believes that 1.5 percent of adults receiving a new credit card offer in the mail will accept it. If the company is correct, what is the probability that less than 1 percent of adults in a random sample of 400 adults receiving the offer will accept it?
A
\(P(z < \dfrac{0.01-0.015}{\sqrt{400(0.01)(0.99)}})\)
B
\(P(z < \dfrac{0.01-0.015}{\sqrt{400(0.015)(0.985)}})\)
C
\(P(z < \dfrac{0.01-0.015}{\sqrt{\dfrac{(0.01)(0.99)}{400}}})\)
D
\(P(z < \dfrac{0.01-0.015}{\sqrt{\dfrac{(0.015)(0.985)}{400}}})\)
E
\(P(z < \dfrac{0.01-0.015}{\sqrt{\dfrac{(0.5)(0.5)}{400}}})\)
27 Probability > Sampling distributions · Level 3
Suppose that the average age of MLB players is 26.8 with a standard deviation of 3.8, while the average age of NFL players is 25.6 with a standard deviation of 2.9. In random samples of 35 MLB players and 30 NFL players, what is the probability that the mean age of the MLB players is greater than the mean age of the NFL players?
A
0.8158
B
0.8324
C
0.9253
D
0.9584
E
0.9860
28 Probability > Sampling distributions · Level 3
Child development scientists gathered data from 65 randomly selected babies and calculated the mean age in weeks at which the babies began to crawl. Is the sample mean \(\overline{x}\) an unbiased estimator for \(\mu\), the mean age at which all babies begin to crawl?
A
No, because some sample means do not equal the population mean
B
No, because there is no reason to assume that the ages at which all babies begin to crawl is normally distributed
C
No, not unless the sample size \(n\) is very large
D
No, not unless the population standard deviation is known
E
Yes, because for random samples, the mean of the sample means is the population mean
29 Probability > Sampling distributions · Level 3
Which of the following statements is incorrect?
A
Like the normal, t-distributions are always symmetric.
B
Like the normal, t-distributions are always bell-shaped.
C
Like the normal, t-distributions are always unimodal.
D
The t-distributions have less spread than the normal, that is, they have less probability in the tails and more in the center than the normal.
E
For larger values of df, degrees of freedom, the t-distributions look more like the normal distribution.
30 Probability > Sampling distributions · Level 3
Which of the following statements about the chi-square distribution is incorrect?
A
There is a separate \(\chi^2\) curve for each df value
B
The area under every \(\chi^2\) curve is the same.
C
For small df, the distribution is skewed to the right; however, for large df, it becomes more symmetric and bell-shaped.
D
For 1 or 2 degrees of freedom, the histogram peak occurs at 0. For 3 or more degrees of freedom, the peak is at df - 2.
E
Just like for the t-distribution, the degrees of freedom for \(\chi^2\) distributions depend upon the sample size.
31 Probability > Sampling distributions · Level 3
Which of the following is a true statement about t-distributions?
A
The greater the number of degrees of freedom, the narrower the tails.
B
The smaller the number of degrees of freedom, the closer the curve is to the normal curve.
C
Thirty degrees of freedom gives the normal curve.
D
The shape of a t-distribution depends on the degrees of freedom, which depend on the population size.
E
The probability that \(z > 1.96\) in a normal distribution is greater than the probability that \(t > 1.96\) in a t-distribution with df = 30.

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