Statistics 101
Exam 3
April 17, 2002
Name:
ID#
INSTRUCTIONS: Read the questions carefully and completely. Answer each question and show
work in the space provided. Partial credit will not be given if work is not shown. When asked to
explain, describe, or comment, do so within the context of the problem.
1. [10 pts] Below is a picture of 20 confidence intervals (vertical lines) constructed from random samples taken from the same population. The population mean is represented by the
horizontal line.
(a) [2] What is the sample mean for the first interval?
(b) [4] What confidence level was used to create these intervals? Explain briefly.
(c) [4] Give an interpretation of what it means to have 95% confidence. In your interpretation you may not use the words: chance, confident, confidence, probability or sure.
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2. [15 pts] For each of the following scenarios indicate what procedure you would use, a zprocedure using Table A or a t-procedure using Table C. Give the formula, a formula contains
no numbers, you would use. You DO NOT need to actually go through the procedure.
(a) [5] A psychologist wishes to construct a 95% confidence interval for the mean IQ score for
all students identified as talented and gifted. A random sample of 25 students identified
as talented and gifted has a mean of 122 with a standard deviation of 8.
(b) [5] A construction engineer wishes to know the mean strength of concrete slabs made
from a new formulation of the concrete. She tests the hypothesis that the mean strength
of the new formulation is 400 psi versus the alternative that the mean is greater than
400 psi. She has data on the strength of 8 slabs made with the new formulation. For
those 8 slabs, X = 452 psi and s = 74.
(c) [5] An exercise physiologist wishes to construct a 95% confidence interval on the mean
amount of lactic acid built up in the muscles after 30 minutes of a modified aerobics
program. He has blood lactic acid levels for a random sample of 16 people. It is believed
that the population of these levels is normally distributed with a standard deviation of
0.014.
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3. [30 pts] Shoshoni Indians of America made beaded rectangles that they used to decorate
their leather goods. An anthropologist is interested in seeing if the Shoshoni Indians used
the idea of the Golden Rectangle (an aesthetic standard for the Egyptians and Greeks) in
making their bead work. The Golden Rectangle has a width-to-length ratio of 0.618, Below
are width-to-length ratios, and summary data, for a random sample of 10 Shoshoni beaded
rectangles.
0.569, 0.606, 0.637, 0.641, 0.662, 0.672, 0.690, 0.693, 0.721, 0.749
X
X = 6.64
X
(X − X)2 = 0.02571
(a) [5] Compute the sample mean, X and sample standard deviation, s for these data.
(b) [2] We wish to test to see if the mean ratio of the population of Shoshoni beaded rectangles is 0.618 against the alternative that this population mean is not 0.618. Give the
null and alternative hypothesis for this test.
(c) [5] Calculate the value of the test statistic and find the P-value.
(d) [2] Assuming 0.05 is a small probability, what is your decision and reason for reaching
that decision?
(e) [3] Based on these data, did the Shoshoni Indians use the Golden Rectangle in making
their bead work? Explain briefly.
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(f) [5] Another way to summarize these data is with a confidence interval. Construct a 95%
confidence interval for the population mean ratio of Shoshoni rectangle beads.
(g) [3] Using the interval in (f), is 0.618 an acceptable value for the population mean ratio?
Explain briefly.
(h) [5] Below is a box plot of the data. Describe the shape of the box plot. Does this support
the condition that data should come from a normally distributed population? Explain
briefly.
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4. [5 pts] Based on a random sample from a normally distributed population with σ = 8, the
90% margin of error is 4.16.
(a) [2] What is the size of the sample taken?
(b) [3] If we want a margin of error of at most 2 with 90% confidence, how many additional
observations do we need?
5. [15 pts] In the past, the mean ACT score for all Freshman enrolled at a certain university
is 21.1 The population standard deviation is 4.5. The university admissions office starts
a program intended to attract students with higher ACT scores. A random sample of 25
Freshmen enrolled for next fall yields a mean ACT of 22.4.
(a) [12] Is this sufficient evidence to conclude that the incoming Freshmen have a mean
ACT significantly higher than before? Provide a complete test of hypothesis.
(b) [3] Suppose there was sufficient evidence to conclude that the incoming Freshmen have a
significantly higher mean ACT (not necessarily the correct answer for part a), could we
attribute that increase to the program started by the admissions office? Explain briefly.
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