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AP Statistics / Mr. Hansen |
Name: __________KEY___________ |
Quiz
on Chapter 10 and Recent Class Discussions
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1. |
A sampling distribution is the set of all possible values
that a statistic can have when samples of a fixed size are drawn from a fixed
population. Although we could consider sampling distributions for
many statistics (s, Q1, range, IQR, etc.), the two statistics of
greatest interest to us in sampling distributions are xbar (the sample mean) and phat
(the sample proportion ). |
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2. |
If a large dataset has a
mean of 83 and a s.d. of 5, then the sample means,
using samples of size 55, will have a mean (or expected value) of
83 and a s.d.
[more correctly, a standard error] of .674. |
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3. |
If 35% of a large
population are “yeses” and 65% are “no’s,” then the sample proportion of “yeses,”
using samples of size 55, will have a mean of .35 and a s.d. of Ö(pq/n) = .064 [which is
technically the s.e.]. The count
of “yeses” in samples of size 55 will have a mean of np
= 55(.35) = 19.25 and a s.d. of Ö(npq) = 3.537 [which,
again, is technically the s.e.]. |
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4. |
In an SRS of 1350 American
voters, 713 give President Bush a “positive” job approval rating. Compute a
95% confidence interval for the true proportion of American voters who feel
this way. No work is required, but state your answer
as a complete sentence using the “approved wording” that we discussed in
class. |
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Do this
by pushing buttons on your calculator: STAT TESTS A (1-PropZInt). |
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For full
credit: “We are 95% confident that the true proportion of American voters who
feel this way is between 50.152% and 55.478%.” |
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OR |
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“We are
95% confident that the true proportion of American voters who feel this way
is 52.815% ± 2.663%. |
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5. |
We measure the heights of
1000 randomly selected American women. A 90% confidence interval for the mean
height of American women is 65 ± .13 inches. Which of the following are true? (Write
the word “YES” in front of each true statement.) |
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The probability is .9 that
the true mean height of American women lies in the
interval between 64.87 and 65.13 inches. |
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The probability is .9 that
the true mean height of American women lies in the
interval from 64.87 through 65.13 inches, inclusive. |
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If we performed the
randomization and measurement procedures again, there is a probability of .9
that the true mean height of American women would fall in the interval from
64.87 to 65.13. The issue of “inclusive” or “non-inclusive” is irrelevant
since the r.v. is continuous. |
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YES |
If we performed the
randomization and measurement procedures again and computed a new confidence
interval, there is a probability of .9 that the true mean height of American
women would fall in the new interval. The issue of “inclusive” or
“non-inclusive” is irrelevant since the r.v. is
continuous. |
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YES |
We are 90% confident that the
true mean height of American women is 65 ± .13 inches. |
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Approximately 90% of
American women have heights between 64.87 and 65.13 inches. |
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6. |
In problem #5, compute the m.o.e. for a 99% confidence level. (Hint: s = 2.5 inches.) This time, show your work. |
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m.o.e. = (crit. val.)(s.e.) = 2.576(s/Ön) = 2.576(2.5/Ö1000) = .204 inches |
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7. |
Mel, who has many tall
female friends, believes based on his anecdotal experience that the estimate
of 65 inches given in problem 5 is too low. State Mel’s null and alternative hypotheses
for the true mean height of American women: |
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(a) |
H0: m = 65 |
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(b) |
Ha:
m > 65 |
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(c) |
Mel will gather his own
large random sample of women and will measure their heights. He will reject H0 if the p value of his test is below .05.
Explain in plain English what a Type I error by Mel would be. |
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concluding
(based on strong evidence in the sample) that the true mean height of
American women exceeds 65 inches, even though the true population mean really
is 65 inches |
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(d) |
State the probability that
Mel makes a Type I error (no work needed). .05 |
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(e) |
Explain in plain English what
a Type II error by Mel would be. |
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(f) |
Explain in plain English
why it is not possible to compute P(Type II error) unless Mel tells you a specific alternative sampling
distribution that he has in mind. |
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Each
specific value of the alternative has a sampling distribution curve (for xbar) associated with it. Only by knowing which curve Mel
has in mind can we calculate how much of that sampling distribution crosses
into the “fail to reject H0” zone, i.e., the Type II error zone. |
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(g) |
Two specific alternatives
Mel is considering are m = 66 and m = 67 inches. Against which of these alternatives does Mel’s
procedure have greater power? Explain briefly. |
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