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AP Statistics Unit 5: Sampling Distributions

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Unit 5: Sampling Distributions

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Question 1

A population is normal with $μ = 100$ and $σ = 15$ . Random samples of size $n = 9$ are repeatedly drawn. What is the shape of the sampling distribution of $\overset{x}{ˉ}$ ?

A.Approximately normal, by the CLT since

n = 9 \geq 5

.B.Exactly normal, because the population is normal.C.Unknown shape, since

n < 30

.D.Approximately normal with mean 100/9.

Question 2

A population has mean $μ = 50$ and standard deviation $σ = 10$ . A simple random sample of size $n = 25$ is taken. What is the standard deviation of the sampling distribution of $\overset{x}{ˉ}$ ?

A.10B.2C.0.4D.50

Question 3

A researcher takes random samples of size $n = 64$ from a population with $μ = 80$ and $σ = 16$ . What is the probability that a sample mean exceeds 82?

P (Z > 2.0) \approx 0.023

P (Z > 1.0) \approx 0.159

P (Z > 0.125) \approx 0.450

P (Z > 4.0) \approx 0.00003

Question 4

A sample proportion $\overset{p}{^} = 0.55$ is calculated from a sample of $n = 200$ . The population proportion is claimed to be $p = 0.50$ . What is the approximate $z$ -score for $\overset{p}{^}$ ?

z \approx 1.41

z \approx 2.00

z \approx 0.14

z \approx 3.16

Question 5

Which statement best describes what the Central Limit Theorem (CLT) tells us about the sampling distribution of $\overset{x}{ˉ}$ ?

A.The population must be normally distributed for

\overset{x}{ˉ}

to be approximately normal.B.For large enough sample sizes, the sampling distribution of

\overset{x}{ˉ}

is approximately normal regardless of the population shape.C.For large samples, the sample mean equals the population mean.D.As sample size increases,

\overset{x}{ˉ}

converges to

σ

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Question 6

A sample of $n = 50$ is taken from a large population where $p = 0.20$ . A student checks the conditions for using a normal approximation and finds $n p = 10$ , $n (1 - p) = 40$ . The student concludes the approximation is barely acceptable. Is this correct?

A.No; the condition requires

n p > 10

strictly, so

n p = 10

fails.B.Yes; both

n p \geq 10

and

n (1 - p) \geq 10

are satisfied, so the normal approximation is justified.C.No; the condition also requires

n \geq 30

, which fails here.D.No; for a proportion, the sample size must be at least 100.

Question 7

A population proportion is $p = 0.40$ . A random sample of $n = 100$ is drawn. What is the standard deviation of the sampling distribution of $\overset{p}{^}$ ?

A.0.40B.0.0024C.0.049D.0.490

Question 8

A population of SAT scores is strongly right-skewed. Samples of size $n = 5$ are taken repeatedly. Which of the following best describes the sampling distribution of $\overset{x}{ˉ}$ for $n = 5$ ?

A.Approximately normal, because the CLT applies for any sample size.B.Exactly normal, because all sampling distributions of

\overset{x}{ˉ}

are normal.C.Still right-skewed, but less skewed than the population.D.Identical in shape to the population distribution.

Question 9

Which condition must be met for the sampling distribution of $\overset{p}{^}$ to be approximately normal?

n \geq 30

n p \geq 10

and

n (1 - p) \geq 10

p \geq 0.5

D.The population must be normally distributed.

Question 10

Two sampling distributions are described: (I) $n = 25$ samples from a symmetric, bell-shaped population; (II) $n = 100$ samples from a heavily skewed population. Which is more likely to be approximately normal?

A.Distribution I, because the population is symmetric.B.Distribution II, because

n = 100

is large enough by the CLT even for skewed populations.C.Both are equally normal, because the CLT applies in both cases.D.Neither is approximately normal, because neither comes from a perfectly normal population.

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Question 11

A factory produces bolts with a mean diameter of $μ = 10$ mm and $σ = 0.5$ mm. Quality control selects random samples of 16 bolts. In repeated sampling, 95% of sample means will fall within what range?

A.9.00 mm to 11.00 mmB.9.75 mm to 10.25 mmC.9.50 mm to 10.50 mmD.9.875 mm to 10.125 mm

Question 12

A large population has proportion $p = 0.60$ with a certain characteristic. A random sample of $n = 100$ is taken. What is the mean of the sampling distribution of $\overset{p}{^}$ ?

A.0.060B.60C.0.60D.0.049

AP Statistics Unit 5: Sampling Distributions — Free with Answer Explanations | Test Practice Hub