1. If z is a standard normal variable, find the probability.
P(-0.73 < z < 2.27)
2. Assume that inspection of a large collection of laboratory thermometers show that at the freezing point of water some give readings below 0 degrees Celsius (denoted by negative numbers) and some give readings above 0 degrees Celsius (denoted by positive numbers). Assume that the mean reading is 0 degrees Celsius and the standard deviation of the readings is 1.00 degrees Celsius.
A quality control analyst wants to examine thermometers that give readings in the bottom 4%. Find the reading that separates the bottom 4% from the others.
-1.75 degrees Celsius
-1.63 degrees Celsius
-1.48 degrees Celsius
-1.89 degrees Celsius
3. Assume that x has a normal distribution, and find the indicated probability.
The true mean, mu = 137.0, and the true standard deviation, sigma = 5.3.
Find the probability that x is between 134.4 and 140.1.
4. The weights of certain machine components are normally distributed with a mean of 8.92 grams and a standard deviation of 0.06 grams. Find the two weights that separate the top 3% and the bottom 3%. These weights could serve as limits used to identify which components should be rejected.
8.89 grams and
8.79 grams and
8.91 grams and
8.81 grams and
5. A bank's loan officer rates applicants for credit. The ratings are normally distributed with a mean of 200 and a standard deviation of 50. If an applicant is randomly selected, find the probability of a rating that is between 170 and 220.
6. A bank's loan officer rates applicants for credit. The ratings are normally distributed with a mean of 200 and a standard deviation of 50. If 40 different applicants are randomly selected, find the probability that their mean is above 215.
7. A study of the amount of time it takes a mechanic to rebuild the transmission for a 1995 Chevrolet Cavalier shows that the mean is 8.4 hours and the standard deviation is 1.8 hours. If 40 mechanics are randomly selected, find the probability that their mean rebuild time is less than 7.6 hours.
8. Supppose the replacement times for washing machines are normally distributed with a mean of 9.3 years and a standard deviation of 1.1 years. Find the probability that 70 randomly selected washing machines will have a mean replacement time less than 9.1 years.
9. A final exam in Math 160 has a mean of 73 with standard deviation 7.8. If 24 students are randomly selected, find the probability that the mean of their test scores is less than 76
10. The amount of snowfall falling in a certain mountain range is normally distributed with a mean of 109 inches, and a standard deviation of 10 inches. What is the probability that the mean annual snowfall during 25 randomly picked years will exceed 111.8 inches?
11. Find the critical value, z alpha-half, that corresponds to a degree of confidence of 98%.
12. A poll of 1400 randomly selected Americans was conducted and found that 30% prefer online banking to traditional banking. Would confidence in the results increase if the sample size were 3200 instead of 1400? Why or why not?
No, the confidence level is set at the beginning of the research and cannot be changed by sample size.
Given the information in the problem, it is not possible to make a definitive statement about confidence.
Yes. A larger sample size will guarantee that the point estimate of the true percentage will virtually equal the parameter.
Yes. As the sample size increases, the sample statistics tend to vary less and they tend to be closer to the population parameter.
13. Express the confidence interval in the form p-hat +/- E.
0.033 < p < 0.493
0.23 +/- 0.5
0.263 +/- 0.23
0.263 +/- 0.5
0.23 +/- 0.6
14. Find the margin of error for the 95% confidence interval used to estimate the population proportion.
n = 163, x = 96
15. Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p.
n = 165, x = 138; 95 percent
0.779 < p < 0.892
0.791 < p < 0.881
0.790 < p < 0.882
0.780 < P < 0.893
16. Find the minimum sample size you should use to assure that your estimate of p-hat will be within the required margin of error around the population p.
Margin of error: 0.04, confidence level: 99%; from a prior study, p-hat is estimated by 0.13.
17. 459 randomly selected light bulbs were tested in a laboratory, 291 lasted more than 500 hours. Find a point estimate of the true proportion of all light bulbs that last more than 500 hours.
18. Use the given degree of confidence and the sample data to construct a confidence interval for the population proportion p.
Of 139 adults selected randomly from one town, 30 of them smoke. Construct a 99% confidence interval for the true percentage of all adults in the town that smoke.
15.8% < p < 27.3%
12.6% < p < 30.6%
13.5% < p < 29.7%
14.7% < p < 28.4%
19. A researcher is interested in estimating the proportion of voters who favor a tax on e-commerce. Based on a sample of 250 people, she obtains the following 99% confidence interval for the population proportion p:
0.113 < p < 0.171.
Which of the statements below is a valid interpretation of this confidence interval?
There is a 99% chance that the true value of p lies between 0.113 and 0.171.
If many different samples of size 250 were selected and, based on each sample, a confidence interval were constructed, 99% of the time the true value of p would lie between 0.113 and 0.171.
If many different samples of size 250 were selected and, based on each sample, a confidence interval were constructed, in the long run 99% of the confidence intervals would contain the true value of p.
If 100 different samples of size 250 were selected and, based on each sample, a confidence interval were constructed, exactly 99 of these confidence intervals would contain the true value of p.
20. Use the confidence level and sample data to find the margin of error, E, for the population mean, mu.
College students' annual earnings: 99% confidence; n = 74, x-bar = $3967, sigma = $874.
21. Use the confidence level and sample data to find a confidence interval for estimating the population mu.
A group of 56 randomly selected students have a mean score of 30.8 with standard deviation of 4.5 on a placement test. What is the 90 percent confidence interval for the mean score, mu, of all students taking the test?
29.2 < mu < 32.4
29.4 < mu < 32.2
29.6 < mu < 32.0
29.8 < mu < 31.8
22. Use the margin of error, confidence level, and standard deviation sigma to find the minimum sample size required to estimate an unknown population mean mu.
Margin of error: $139, confidence level: 95%, sigma = $513
23. Use the given degree of confidence and sample data to construct a confidence interval for the population mean mu. Assume that the population has a normal distribution.
A savings and loan association needs information concerning the checking account balances of its local customers. A random sample of 14 accounts was checked and yielded a mean balance of $664.14 and a standard deviation of $297.29. Find a 98% confidence interval for the true mean checking account balance for local customers.
$492.52 < mu < $835.76
$493.71 < mu < $834.57
$453.59 < mu < $874.69
$455.65 < mu < $872.63
24. A researcher wishes to construct a 95% confidence interval for a population mean. She selects a simple random sample of size n = 20 from the population. The population is normally distributed and sigma is unknown. When constructing the confidence interval, the researcher should use the t distribution; however, she incorrectly uses the normal distribution. Which of the following describes her result?
The true confidence level of the resulting confidence interval is smaller than 95%.
The true confidence level of the resulting confidence interval is greater than 95%.
The true confidence level of the resulting confidence interval is exactly 95%.
There is not enough information in the problem to make an accurate judgment.
25. Use the given degree of confidence and sample data to find a confidence interval for the population standard deviation sigma. Assume that the population has a normal distribution.
The mean replacement time for a random sample of 20 washing machines is 9.5 years and the standard deviation is 2.4 years. Construct a 99% confidence interval for the standard deviation, sigma, of the replacement times for all washing machines of this type.
1.7 yr <sigma< 3.8 yr
1.6 yr <sigma< 4.6 yr
1.7 yr <sigma< 4.0 yr
1.7 yr <sigma< 5.0 yr