Part A - PROBABILITY - Maximum and Minimum Temperatures
Search the Internet for U.S. climate data.
Choose the city in which you live. (Somerset NJ 08873)
Click on the tab that reads "Daily."
1. Prepare a spreadsheet with three columns: Date, High Temperature, and Low Temperature. List the past 60 days for which data is available.
2. Prepare a histogram for the data on high temperatures and comment on the shape of the distribution as observed from these graphs.
3. Calculate X- and S.
4. What percentage of the high temperatures are within the interval X- - S to X- + S?
5. What percentage of the high temperatures are within the interval X- - 2S to X- + 2S?
6. How do these percentages compare to the corresponding percentages for a normal distribution (68.26% and 95.44%, respectively)?
7. Repeat Parts 2 to 6 for the minimum temperatures on your spreadsheet.
8. Would you conclude that the two distributions are normally distributed? Why or why not?
Part B - INTRODUCTION TO STATISTICAL THINKING
Directions: Complete the following questions. The most important part of statistics is the thought process, so make sure that you explain your answers, but be careful with statistics. The following statistics/probability problems may intrigue you and you may be surprised. The answers are not always as you might think. Please answer them as well as you can by using common logic.
1. There are 23 people at a party. Explain what the probability is that any two of them share the same birthday.
2. A cold and flu study is looking at how two different medications work on sore throats and fever. Results are as follows:
- Sore throat - Medication A: Success rate - 90% (101 out of 112 trials were successful)
- Sore throat - Medication B: Success rate - 83% (252 out of 305 trials were successful)
- Fever - Medication A: Success rate - 71% (205 out of 288 trials were successful)
- Fever - Medication B: Success rate - 68% (65 out of 95 trials were successful)
Analyze the data and explain which one would be the better medication for both a sore throat and a fever.
3. The United States employed a statistician to examine damaged planes returning from bombing missions over Germany in World War II. He found that the number of returned planes that had damage to the fuselage was far greater than those that had damage to the engines. His recommendation was to enhance the reinforcement of the engines rather than the fuselages. If damage to the fuselage was far more common, explain why he made this recommendation.
Part C -
Q1. The director of admissions at Kinzua University in Nova Scotia estimated the distribution of student admissions for the fall semester on the basis of past experience.
|
Admissions
|
Probability
|
|
1,090
|
0.4
|
|
1,340
|
0.3
|
|
1,640
|
0.3
|
|
1. What is the expected number of admissions for the fall semester?
2. Compute the variance and the standard deviation of the number of admissions.
Q2. Customers experiencing technical difficulty with their internet cable hookup may call an 800 number for technical support. It takes the technician between 90 seconds and 14 minutes to resolve the problem. The distribution of this support time follows the uniform distribution.
a. What are the values for a and b in minutes?
b-1. What is the mean time to resolve the problem?
b-2. What is the standard deviation of the time?
c. What percent of the problems take more than 5 minutes to resolve?
d. Suppose we wish to find the middle 50% of the problem-solving times. What are the end points of these two times?
Q3. A normal population has a mean of 18 and a standard deviation of 4.
a. Compute the z value associated with 23.
b. What proportion of the population is between 18 and 23?
c. What proportion of the population is less than 15?
Q4. Assume that the hourly cost to operate a commercial airplane follows the normal distribution with a mean of $2,100 per hour and a standard deviation of $250.
What is the operating cost for the lowest 3% of the airplanes? (Round z value to 2 decimal places and round final answer to nearest whole dollar.)
Q5. The manufacturer of a laser printer reports the mean number of pages a cartridge will print before it needs replacing is 12,425. The distribution of pages printed per cartridge closely follows the normal probability distribution and the standard deviation is 595 pages. The manufacturer wants to provide guidelines to potential customers as to how long they can expect a cartridge to last.
How many pages should the manufacturer advertise for each cartridge if it wants to be correct 99 percent of the time?
Q6. A study of long-distance phone calls made from General Electric Corporate Headquarters in Fairfield, Connecticut, revealed the length of the calls, in minutes, follows the normal probability distribution. The mean length of time per call was 4.20 minutes and the standard deviation was 0.40 minutes.
a. What fraction of the calls last between 4.20 and 4.90 minutes?
b. What fraction of the calls last more than 4.90 minutes?
c. What fraction of the calls last between 4.90 and 5.50 minutes?
d. What fraction of the calls last between 3.50 and 5.50 minutes?
e. As part of her report to the president, the director of communications would like to report the length of the longest (in duration) 6 percent of the calls. What is this time?
Q7. A population consists of the following five values: 12, 14, 15, 16, and 17.
a. List all samples of size 3, and compute the mean of each sample.
b. Compute the mean of the distribution of sample means and the population mean.
Q8. The mean age at which men in the United States marry for the first time follows the normal distribution with a mean of 24.6 years. The standard deviation of the distribution is 2.9 years.
For a random sample of 64 men, what is the likelihood that the age at which they were married for the first time is less than 24.9 years?
Part D -
Q1. A sample of 32 observations is selected from a normal population. The sample mean is 32, and the population standard deviation is 6. Conduct the following test of hypothesis using the 0.05 significance level.
H0: μ ≤ 29
H1: μ > 29
a. Is this a one- or two-tailed test?
"One-tailed"-the alternate hypothesis is greater than direction.
"Two-tailed"-the alternate hypothesis is different from direction.
b. What is the decision rule?
c. What is the value of the test statistic?
d. What is your decision regarding H0?
e. What is the p-value?
Q2. At the time she was hired as a server at the Grumney Family Restaurant, Beth Brigden was told, "You can average $71 a day in tips." Assume the population of daily tips is normally distributed with a standard deviation of $4.48. Over the first 33 days she was employed at the restaurant, the mean daily amount of her tips was $73.33. At the 0.05 significance level, can Ms. Brigden conclude that her daily tips average more than $71?
a. State the null hypothesis and the alternate hypothesis.
H0: μ ≤ 71; H1: μ > 71
H0: μ ≥ 71; H1: μ < 71
H0: μ >71; H1: μ = 71
H0: μ = 71; H1: μ ≠ 71
b. State the decision rule.
Reject H1 if z > 1.65
Reject H1 if z < 1.65
Reject H0 if z > 1.65
Reject H0 if z < 1.65
c. Compute the value of the test statistic.
d. What is your decision regarding H0?
e. What is the p-value?
Q3. The Rocky Mountain district sales manager of Rath Publishing Inc., a college textbook publishing company, claims that the sales representatives make an average of 37 sales calls per week on professors. Several reps say that this estimate is too low. To investigate, a random sample of 43 sales representatives reveals that the mean number of calls made last week was 38. The standard deviation of the sample is 2.6 calls. Using the 0.010 significance level, can we conclude that the mean number of calls per salesperson per week is more than 37?
H0: μ ≤ 37
H1: μ > 37
1. Compute the value of the test statistic.
2. What is your decision regarding H0?
Q4. A United Nations report shows the mean family income for Mexican migrants to the United States is $26,450 per year. A FLOC (Farm Labor Organizing Committee) evaluation of 23 Mexican family units reveals a mean to be $37,190 with a sample standard deviation of $10,700. Does this information disagree with the United Nations report? Apply the 0.01 significance level.
a. State the null hypothesis and the alternate hypothesis.
b. State the decision rule for .01 significance level.
c. Compute the value of the test statistic.
d. Does this information disagree with the United Nations report? Apply the 0.01 significance level.
Q5. The following information is available.
H0: μ ≥ 220
H1: μ < 220
A sample of 64 observations is selected from a normal population. The sample mean is 215, and the population standard deviation is 15. Conduct the following test of hypothesis using the .025 significance level.
a. Is this a one- or two-tailed test?
b. What is the decision rule?
c. What is the value of the test statistic?
d. What is your decision regarding H0?
e. What is the p-value?
Q6. Given the following hypotheses:
H0: μ ≤ 10
H1: μ > 10
A random sample of 10 observations is selected from a normal population. The sample mean was 12 and the sample standard deviation 3. Using the .05 significance level:
a. State the decision rule.
b. Compute the value of the test statistic.
c. What is your decision regarding the null hypothesis?
Q7. Given the following hypotheses:
H0: μ = 400
H1: μ ≠ 400
A random sample of 12 observations is selected from a normal population. The sample mean was 407 and the sample standard deviation 6. Using the .01 significance level:
a. State the decision rule.
b. Compute the value of the test statistic.
c. What is your decision regarding the null hypothesis?