Download Assignment 4: Quiz

Part I T/F & Multiple Choice
1. For a specific confidence interval, the larger the sample size, the smaller the maximum error of estimate will be. _____
T/F
2. An estimator is consistent if, as the sample size decreases, the value of the estimator approaches the value of the
parameter estimated. ____ T/F
3. When we reject the null hypothesis, we are certain that the null hypothesis is false. ___ T/F
4. The alternative hypothesis, sometimes referred to as the research hypothesis, is supported by using the sample
evidence to contradict the null hypothesis. ___ T/F
5. Rejection of a null hypothesis that is false is a Type II error. ____ T/F
6.  is the measure of the area under the curve of the standard score that lies in the rejection region for the null
hypothesis. ___ T/F
7. The maximum error of estimate is controlled by three factors: level of confidence, sample size, and standard deviation.
____ T/F
8. You are constructing a 95% confidence interval using the information: n = 60, x = 65.5, s = 2.5, and E = 0.7. What is
the value of the middle of the interval?
A. 0.7
B. 2.5
C. 0.95
D. 65.5
9. Which of the following would be the correct hypotheses for testing the claim that the mean life of a battery for a cellular
phone (while the phone is left on) is less than 24 hours?
A. H0:   24 and Ha:  < 24
B. H0:  = 24 and Ha:   24
C. H0:   24 vs. Ha:  > 24
D. H0:  > 24 vs. Ha:   24
10. Which of the following would be the alternative hypothesis in testing the claim that the mean distance students
commute to campus is no more than 8.2 miles?
A. Ha:   8.2
B. Ha:  > 8.2
C. Ha:  < 8.2
D. Ha:   8.2
Part II. Short Answers & Computational Questions
1. A study was conducted to estimate the mean amount spent on birthday gifts for a typical family having two children. A
sample of 150 was taken, and the mean amount spent was $225. Assuming a standard deviation equal to $50, find the
95% confidence interval for , the mean for all such families.
SE = s/n = 50/150 = 4.082
The 95% CI for the mean amount spent is given by [M – 1.96 * SE, M + 1.96 * SE]
= [225 – 1.96 * 4.082, 225 + 1.96 * 4.082]
= [$217, $233]
2. What sample size would be needed to estimate the population mean to within one-half standard deviation with 95%
confidence?
n = (z * s/Margin of error)^2 = [1.96 * s/(0.5 * s)]^2 = 15.37
A sample size of 16 is required.
3. A 95% confidence interval estimate for a population mean was computed to be (44.8 to 50.2). Determine the mean of
the sample, which was used to determine the interval estimate.
Mean = (44.8 + 50.2)/2 = 47.5
4. In testing the hypothesis, H0:   28.7 and Ha:  < 28.7, using the p-value approach, a p-value of 0.0764 was obtained.
If  = 9.8, find the sample mean which produced this p-value given that the sample of size n = 40 was randomly selected.
The z- score corresponding to a p- value (Left-tailed test) of 0.0764 is -1.43
x-bar =  + z * SE = 28.7 + (-1.43)(9.8/40) = 26.48
5. To test the null hypothesis that the average lifetime for a particular brand of bulb is 750 hours versus the alternative
that the average lifetime is different from 750 hours, a sample of 75 bulbs is used. If the standard deviation is 50 hours
and  is equal to 0.01, what values for x will result in rejection of the null hypothesis.
SE = s/n = 50/75 = 5.77
The critical value of z corresponding to  = 0.01 (Two-tailed) is 2.576
z = (x-bar – )/SE
To reject the null hypothesis, we need (x-bar – 750)/5.77 < -2.576 or (x-bar – 750)/5.77 > 2.576
x-bar < 750 - 5.77 * 2.576 or x-bar > 750 + 5.77 * 2.576
x-bar < 735.14 hours or x-bar > 764.86 hours
6. Describe the action that would result in a type I error and a type II error if each of the following null hypotheses were
tested.
a. H0: There is no waste in US Defense Department spending
b. H0: This fast-food menu is not low fat
(a) If there really is no waste in US Defense Department spending and if H0 is rejected then a Type I error is
committed. If there is waste in US Defense Department spending and if we fail to reject H0 then a Type II
error is committed.
(b) If the fast food menu is really not low fat and if H0 is rejected then a Type I error is committed. If the fast
food menu is low fat and if we fail to reject H0 then a Type II error is committed.
7. By measuring the amount of time it takes a component of a product to move from one workstation to the next, an
engineer has estimate that the standard deviation is 4.5 seconds.
a. How many measurements should be made in order to be 95% certain that the maximum error of estimation will not
exceed 1 second?
b. What sample size is required for a maximum error of 2 seconds?
(a) n = (z * s/Margin of error)^2 = [1.96 * 4.5/1]^2 = 78
A sample size of 78 is required.
(b) n = (z * s/Margin of error)^2 = [1.96 * 4.5/2]^2 = 20
A sample size of 20 is required.
8. Determine the critical region and critical values for z that would be used to test the null hypothesis at the given level of
significance, as described in each of the following:
a. H0:  = 25 and Ha:   25,  = 0.10
b. H0:   32 and Ha:  > 32,  = 0.01
c. H0:   13 and Ha:  < 13,  = 0.05
(a) z = ± 1.645, critical region is the region in the two tails
(b) z = 2.326, critical region is the region in the right tail
(c) z = -1.645, critical region is the region in the left tail.