Download AP Statistics Name Chapter 7 Review KEY Suppose you take a

AP Statistics
Name ____________________
Chapter 7 Review KEY
1. Suppose you take a random sample of size 25 from a population with mean of 110 and a
standard deviation of 25. Your sample has a mean of 115 and a standard deviation of 23.8.
Which of the following has a mean of 110 and a standard deviation of 25?
A The distribution of the population – if it has a standard deviation of 25, then it can’t be
the sample data or sampling distribution.
2. You take a sample of size 30 from a very large population in which the true proportion is p = 0.2,
thus violating the condition that 𝑛𝑝 ≥ 10 and 𝑛(1 − 𝑝) ≥ 10. Which statement below best
describes what you know about the sampling distribution of 𝑝̂ ?
𝑫
(0.2)(0.8)
;
30
𝜇𝑝̂ = 0.2; we cannot use the formula 𝜎𝑝̂ = √
the distribution is not
approximately Normal. – If it violates the condition above, then we cannot use the
standard deviation formula and the distribution is not Normal.
3. You take an SRS of size 100 from the 37,000 students at Chico State University and measure
individual’s heights. You then take an SRS of size 100 from the 6,500,000 adults in the state of
California and measure their heights. Assuming the standard deviation of individual heights in
the two populations is the same, the standard deviation of the sampling distribution of mean
heights for the California sample is
A
Approximately the same as for the Chico sample because both are samples of
size 100 – If the population standard deviations are the same for the Chico and California
sample, and the sample size is both 100, then their sample standard deviation’s will be
exactly the same given the formula,
.
4. A student investigating study habits asks a simple random sample of 16 students at her school
how many minutes they spent on their English homework the previous night. Suppose the
actual parameter values for this variable are 𝜇 = 50 minutes and 𝜎 = 10 minutes. Which of the
following best describes what we know about the sampling distribution of means for the
student’s sample?
𝑬
𝜇𝑥̅ = 50; 𝜎𝑥̅ = 2.5; shape of distribution unknown
AP Statistics
Name ____________________
Chapter 7 Review KEY
Free Response
5. In a study of the effects of acid rain, a random sample of 10,000 trees from a particular forest is
examined. Forty five percent of the trees show some signs of damage. Define the following in
this scenario:
a) Statistic - 𝑃̂ = .45, proportion of trees showing some signs damage from the SRS.
b) Population – All the trees from that particular forest.
c) Sample – The 10,000 randomly selected trees from that particular forest.
6. Define
a) The sampling distribution of the statistic – the distribution of values taken by the statistic in
all possible samples of the same size from the same population.
b) Unbiased estimator – A statistic used to estimate a parameter if the mean of its sampling
distribution is equal to the value of the parameter being estimated
c) Central Limit Theorem – when n is large, the sampling distribution of proportion or mean
samples is approximately normal
7. Interpupillary distance (IPD) is the distance between the centers of the pupils of a person’s left
and right eyes. In adult males IPD is approximately Normally distributed with a mean of 63.5 mm
and a standard deviation of 5 mm. Suppose you randomly select 5 adult males. What is the
probability that their mean IPD is greater than 60 mm?
𝜎(𝑚𝑒𝑎𝑛) =
𝜎(𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛)
√𝑛
=
5
√5
= 2.24 P(mean > 60) =
60−63.5
2.24
= -1.46
P(Z>-1.46) = 1-.0721 = 92.79%
8. In a large population, 78% of the households have Wifi. A simple random sample of 100
households from this population is to be contacted and the sample proportion computed. What
is the probability that more than 80% the households sampled will have Wifi?
(𝑝)(1−𝑝)
𝑛
𝜎(𝑚𝑒𝑎𝑛) = √
= √
(.78)(1−.78)
100
P(Z>0.48) = 1-.6844 = 31.56%
= 0.0414
.80−.78
P(𝑃̂ > .80) = 0..0414 = 0.48
AP Statistics
Name ____________________
Chapter 7 Review KEY
9. The graph below displays the relative frequency distribution for X, the total number of dogs and
cats owned per household, for the households in a large suburban area. For instance, 14% of the
households own 2 of these pets.
a) According to a local law, each household in this area is prohibited from owning more than 3
of these pets. If a household in this area is selected at random, what is the probability that
the selected household will be in violation of this law?
P(X > 3) = 0.08 + 0.07 + 0.03 + 0.02 = .20
b) If 10 households in this area are selected at random, what is the probability that exactly 2 of
them will be in violation of this law? This was the formula from last chapter how to find the
probability of a binomial distribution (chapter 6).
Y = number of households in violation.
Y has a binomial distribution with n = 10 and p = .20
10
P(y = 2) = ( ) (.20)2 (.80)8 = .3020
2
c) The mean and standard deviation of X are 1.65 and 1.851, respectively. Suppose 150
households in this area are to be selected at random and 𝑋̅, the mean number of dogs and
cats per household, is to be computed. Describe the sampling distribution of 𝑋̅,including its
shape, center and spread.
The distribution of X will: (a) be approximately normal, (b) have mean = 1.65, have a
standard deviation of
𝜎(𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛)
√𝑛
=
1.851
√150
= 0.1511
AP Statistics
Name ____________________
Chapter 7 Review KEY
10. Big Town Fisheries recently stocked a new lake in a city park with 4,000 fish of various sizes. The
distribution of the length of these fish is approximately normal.
a) Big Town Fisheries claims that the mean length of the fish is 8 inches. If the claim is true,
which of the following would be more likely?
 A random sample of 300 fish having a mean length that is greater than 10 inches
or
 A random sample of 30 fish having a mean length that is greater than 10 inches
JUSTIFY YOUR ANSWER
The random sample of n = 30 fish is more likely to have a sample mean length
greater than 10 inches. The sampling distribution of the sample mean is normal with
mean = 8, and a standard deviation of
𝜎(𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛)
.
√𝑛
Thus both sampling
distributions will be centered at 8 inches, but the sampling distribution of the
sample mean when n= 30 will have more variability than the sampling distribution
of the sample mean when n = 300. The tail area (mean >10) will be larger for the
distribution that is less concentrated about the mean of 9 inches when the sample
size is n = 30, as shown in the following graph.
AP Statistics
Name ____________________
Chapter 7 Review KEY
b) Suppose the standard deviation of the sampling distribution of the sample mean for random
samples of size 30 is 0.3 inch. If the mean length of the fish is 8 inches, use the normal
distribution to compute that a random sample of 30 fish will have a mean length less than
7.2 inches. The answer below is for less than 7.5 so it is not the exact answer but plug in
7.2 the same way and you will get the right answer.
c) Suppose the distribution of fish lengths in this lake was non-normal but had the same mean
and standard deviation. Would it still be appropriate to use the normal distribution to
compute the probability in part b? Justify your answer.
Yes. The CLT says that the sampling distribution of the sample mean will become approximately
normal as the sample size n increases. Since the sample size is reasonably large (n≥ 30), the calculation
in part (b) will provide a good approximation to the probability of interest even though the population
is nonnormal.