Download Test 10B

Test 10B
AP Statistics
Name:
Directions: Work on these sheets. Answer completely, but be concise. A normal
probability table is attached.
1. In formulating hypotheses for a statistical test of significance, the null hypothesis is
often
A) a statement that there is “no effect” or “no difference.”
B) the probability of observing the data you actually obtained.
C) a statement that the data are all 0.
D) 0.05.
2. An agricultural researcher plants 25 plots with a new variety of corn. A 90%
confidence
interval for the average yield for these plots is found to be 162.72 ± 4.47 bushels per
acre. Which of the following would produce a confidence interval with a smaller margin
of error than this 90% confidence interval?
A) planting only five plots rather than 25, since five are easier to manage and
control.
B) planting 100 plots rather than 25.
C) computing a 99% confidence interval rather than a 90% confidence interval;
the increase in confidence indicates that we have a better interval.
D) none of the above.
3. A small company consists of 25 employees. As a service to the employees, the
company arranges for each to have a complete physical for free. Among other things,
the weight of each employee is measured; the mean weight is found to be 165 pounds.
The population standard deviation is 20 pounds. It is believed that a mean weight of 160
pounds would be normal for this group. To see if there is evidence that the mean weight
of the population of all employees of the company differs significantly from 160, the
hypotheses H0: μ = 160, Ha: μ > 160 are tested. You obtain a P-value of less than 0.1056.
Which of the following is true?
A) At the 5% significance level, you have proved that H0 is true.
B) You have failed to obtain any evidence for Ha.
C) At the 5% significance level, you have failed to prove that H0 is true, and a
larger sample size is needed to do so.
D) None of the above.
4. The mean area μ of the several thousand apartments in a new development is
advertised to be 1250 square feet. A tenant group thinks that the apartments are
smaller than advertised. They hire an engineer to measure a sample of apartments to
test their suspicion. The appropriate null and alternative hypotheses, H0 and Ha, for μ
A) are H0: μ = 1250, Ha: μ ¹ 1250.
B) are H0: μ = 1250, Ha: μ < 1250.
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C) are H0: μ = 1250, Ha: μ > 1250.
D) cannot be specified without knowing the size of the sample used by the
engineer.
5. An agricultural researcher plants 25 plots with a new variety of corn. The average
yield for these plots is J = 150 bushels per acre. Assume that the yield per acre for the
new variety of corn follows a normal distribution with unknown mean μ and that a 95%
confidence interval for μ is found to be 150 ± 3.29. Which of the following is true?
A) A test of the hypotheses H0: μ = 150, Ha: μ ¹ 150 would be rejected at the
0.05 level.
B) A test of the hypotheses H0: μ = 150, Ha: μ > 150 would be rejected at the
0.05 level.
C) A test of the hypotheses H0: μ = 160, Ha: μ ¹ 160 would be rejected at the
0.05 level.
D) No hypothesis test can be conducted because we do not know s.
6. In tests of significance about an unknown parameter of some population, which of the
following is considered strong evidence against the null hypothesis?
A) The value of an estimate of the unknown parameter based on a simple random
sample from the population is not equal to zero.
B) The value of an estimate of the unknown parameter based on a simple random
sample from the population is equal to zero.
C) We observe a value of an estimate of the unknown parameter based on a simple
random sample from the population that is very consistent with the null
hypothesis.
D) We observe a value of an estimate of the unknown parameter based on a simple
random sample from the population that is very unlikely if the null hypothesis is
true.
7. A procedure for approximating sampling distributions (which can then be used to
construct confidence intervals) when theory cannot tell us their shape is
A) least squares.
B) the bootstrap.
C) residual analysis.
D) standardization.
8. A medical researcher is working on a new treatment for a certain type of cancer. The
average survival time after diagnosis on the standard treatment is two years. In an
early trial, she tries the new treatment on three subjects who have an average survival
time after diagnosis of four years. Although the survival time has doubled, the results
are not statistically significant even at the 0.10 significance level. Suppose, in fact, that
the new treatment does increase the mean survival time in the population of all patients
with this particular type of cancer. The researcher has
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A) committed a type I error.
B) committed a type II error.
C) incorrectly used a level 0.10 test when she should have used a 0.05 level test.
D) incorrectly used a level 0.10 test when she should have computed the P-value.
Use the following to answer question 9:
A researcher wishes to determine if students are able to complete a certain pencil-andpaper maze more quickly while listening to classical music. Suppose the time (in seconds)
needed for high school students to complete the maze while listening to classical music
follows a normal distribution with mean μ and standard deviation s = 4. Suppose also that
in the general population of all high school students the time needed to complete the
maze (without listening to classical music) follows a normal distribution with mean 40
and standard deviation s = 4. The researcher, therefore, decides to test the hypotheses
H0: μ = 40, Ha: μ < 40
To do so, the researcher has 10,000 high school students complete the maze with
classical music playing. The mean time for these students is J = 39.8 seconds and the Pvalue is less than 0.0001.
9. Suppose that two high school students decide to see if they get the same results as
the researcher. They both take the maze while listening to classical music. The mean of
their times is J = 39.8 seconds, the same as that of the researcher. It is appropriate to
conclude which of the following?
A) The students have reproduced the results of the researcher, and their P-value
will be the same as that of the researcher.
B) The students have reproduced the results of the researcher, but their P-value
will be slightly smaller than that of the researcher.
C) The students will reach the same statistical conclusion as the researcher, but
their Pvalue will be a bit different than that of the researcher.
D) None of the above.
10. In a test if statistical hypotheses, the P-value tells us
A) if the null hypothesis is true.
B) if the alternative hypothesis is true.
C) the largest level of significance at which the null hypothesis can be rejected.
D) the smallest level of significance at which the null hypothesis can be rejected.
11. A certain population follows a normal distribution with mean μ and standard deviation
s= 2.5. You collect data and test the hypotheses
H0: μ = 1, Ha: μ ¹ 1
You obtain a P-value of 0.022. Which of the following is true?
A) A 95% confidence interval for μ will include the value 1.
B) A 95% confidence interval for μ will include the value 0.
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C) A 99% confidence interval for μ will include the value 1.
D) A 99% confidence interval for μ will include the value 0.
12. An engineer designs an improved light bulb. The previous design had an average
lifetime of 1200 hours. The mean lifetime of a random sample of 2000 new bulbs is
found to have a mean lifetime of 1201 hours. Although the difference from the old mean
lifetime of 1200 hours is quite small, the P-value is 0.03 and the effect is statistically
significant at the 0.05 level. If, in fact, there is no difference between the mean
lifetimes of the new and old designs, the researcher has
A) committed a type I error.
B) committed a type II error.
C) a probability of being correct that is equal to the P-value.
D) a probability of being correct that is equal to 1 – (P-value).
Part 2:
Free Response
Communicate your thinking clearly and completely.
9. A steel mill’s milling machine produces steel rods that are supposed to be 5 cm in
diameter. When the machine is in statistical control, the rod diameters vary
according to a normal distribution with mean µ = 5 cm and standard deviation
cm. A large sample of 150 produced by the machine yields a sample mean diameter
of 5.005 cm.
(a) Construct a 99% confidence interval for the true mean diameter of the rods
produced by the milling machine. Follow the inference toolbox.
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(b) Does the interval in (a) give you reason to suspect that the machine is not
producing rods of the correct diameter? State appropriate hypotheses and a
significance level. Then explain your conclusion.
(c) Describe a Type II error in the context of this problem. How could the
manufacturer decrease the probability of a Type II error.
10.A pharmaceutical manufacturer does a chemical analysis to check the potency of
products. The standard release potency for cephalothin crystals is 910. An assay of
16 lots gives the following potency data:
897 914 913 906
916 918 905 921
918 906 895 893
908 906 907 901
(a) Construct a 99% confidence interval for the population mean. Follow the
Inference Toolbox.
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You want to test hypotheses about the mean population potency,
H 0 : µ = 910
H a : µ < 910


at the 1% level of significance. The z test statistic is z = x  910 8.2

16 .
(b) What is the rule for rejecting H 0 in terms of z?
(c) What values of x would lead you to reject H 0 ?
(d)Describe a Type I error in the context of this problem. What is the probability
of a Type I error?
I pledge that I have neither given nor received aid on this
test.________________________
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