Review Sheet - Test 3

1. Explain, clearly and briefly, in your own words the following.

  1. confidence interval (p. 298)
  2. level of confidence (p. 299)
  3. critical value (p. 301)
  4. E, margin of error (p. 302)
  5. Student t distribution (p. 313)
  6. statistical hypothesis (p. 366)
  7. Type I error (p. 375)
  8. Type II error (p. 375)
  9. critical region (p. 372)
  10. alpha, significance level (p. 372)
  11. one and two-tailed tests (p. 373)
  12. P-value (p. 387)
  13. scatterdiagram (p. 507)
  14. linear correlation coefficient (p. 509)
  15. positive correlation (p. 508)
  16. negative correlation (p. 508)
  17. regression equation (line of best fit)(p. 525)
  18. method of least squares (p. 533)

2. Explain, clearly and briefly, in your own words the following.

  1. estimating with confidence intervals (Chapter 6)
  2. hypothesis testing (Chapter 7)
  3. Chi Square test for independence (Chapter 10)
  4. Chi Square test for goodness of fit (Chapter 10)
  5. correlation analysis (Chapter 9)
  6. regression analysis (Chapter 9)

3.

  1. When would you use the t-distribution in hypothesis testing or in estimating?
  2. What would you choose to be your level of significance in a hypothesis test? Why?
  3. If you would reject the null hypothesis at a P-value = .3456, how would you report the results of your test? Is it a reasonable idea to reject at this level?

4. Some researchers have found that there is a positive correlation between coffee drinking and the risk of heart disease.

  1. Does this mean that coffee drinking causes heart disease?
  2. What third factor could be causing both factors to increase?

5. A marketing firm wanted to estimate the average amount spent by shoppers entering a small grocery store in midtown. For 130 randomly selected customers, the mean amount spent was $24.50 and the standard deviation was $4.80.

  1. Find a 90% confidence interval for the mean amount spent by shoppers at this store?
  2. Find a 95% confidence interval for this mean
  3. Find a 99% confidence interval for the same mean.
  4. Explain clearly the relationship between confidence level and the width of the confidence interval.

6. At the beginning of the fall semester, a poll showed that 44% of the students in dormitories at a large university wanted loud music turned off at 10 o'clock on nights before school days. Proponents of the music curfew conducted a second poll in November hoping to show that the percentage had increased as exams drew near. 120 students were randomly selected and they found that 63 of them favored the music curfew.

  1. Perform the appropriate hypothesis test at the significance level, σ = .05.
  2. What is the minimum level of significance at which you can reject the current value of 44%?

7. Considering how common international travel has become, you are asked to investigate the number of resident U.S. citizens in a country (x) and the number of U.S. tourists in the same country (y). You use the Statistical Abstract of the United States, 1994 and randomly choose 12 countries for your study. The information you collected is recorded below:

Countryx = number of U.S. citizens who are residents (in thousands)y = number of U.S. tourists (in thousands)
Argentina 13 11
Canada 296 495
Dominican Republic 97 36
France 59 89
Greece 32 60
Ireland 46 86
Italy 104 105
Israel 112 61
Panama 36 2
Saudi Arabia 40 6
Spain 79 155
United Kingdom 255 377
  1. Draw a scatterdiagram for this data. COMMENT.
  2. List the correlation coefficient.
  3. Test the correlation coefficient, r , for significance at the 0.05 level of significance.
  4. If it is appropriate, list the equation of the line of best fit.
  5. Explain clearly, and in sentence form, the meaning of the slope of this line.
  6. Explain clearly, and in sentence form, the meaning of the y-intercept of this line.
  7. Use this equation to predict the number of U.S. tourists in a country with 100 thousand resident U.S. citizens.
  8. Do you believe that there is a cause and effect relationship between x and y ? EXPLAIN.

8. A sociologist wishes to see whether the number of years of college a person has completed is related to his/her place of residence. A sample of 1000 people is selected and classified as given below. Test the hypothesis that the years of college and the residence location are independent. Test at the 5% level of significance.

Location No College College Degree Graduate Degree Total
Urban 150 120 80 350
Suburban 80 150 70 300
Rural 160 90 100 350
Total 390 360 250 1000

9. For a group of 25 men subjected to a stress test, the mean number of heartbeats per minute was 126 and the standard deviation was 4. Assume the measure is approximately normal.

  1. Find a 95% confidence interval for population value for mean number of heartbeats.
  2. How would you have handled this problem if you knew the population standard deviation?
  3. How would you have handled this problem if the sample size was 100 instead of 25?
  4. How would you have handled this problem if the parent distribution was not approximately normal?

10. The table below shows data from the Statistical Abstract of the United States, 1994. It shows the relationship between gender and ethnic groups among Americans of Hispanic descent who participate in the work force. (A group of 14,770,000 workers.) A social scientist is interested in sampling these ethnic groups to see if gender and ethnic group are independent for Hispanic-American workers. His sample results are listed.

Gender Mexican Puerto Rican Cuban Other
Male 81 72 76 82
Female 51 46 53 57
  1. Perform the appropriate test at the 1% level of significance.
  2. Write your conclusion using P-values.

11. A researcher is interested in estimating the average salary of police officers in a city. She wants to be 95% sure that her estimate is correct to within $200 of the true mean salary. A pilot study was conducted to determine an estimate for standard deviation to be $1250. How large a sample will she need to measure the average salary to within $200?

12.

  1. Would the correlation between the age of a used car and its price be positive or negative? EXPLAIN. (Antique cars are not included.)
  2. Suppose men always married women who were exactly 8% shorter than they are. What would the correlation between their heights be?
  3. Would the number of days absent from class and the grade received in the class have a positive, negative or zero correlation? EXPLAIN.

13. If students are to evaluate a class and instructor properly, they need sufficient time to fill out the student evaluation form. One form under consideration has a claim that it takes only 10 minutes to fill out. To determine if the claim is correct, a random sample of 32 students were asked to fill in the form to evaluate their instructor for their first class on Monday. The times for completion in minutes were:

1285129121591191311127910
1013151618121510131215613141011
  1. Test the claim that the average time required to fill out the form is significantly different from 10 minutes. Use a 5% level of significance.
  2. Find a 95% confidence interval for the average time required to fill out the form.

14.

  1. Choose one of the following responses and explain your choice.

    "The correlation between the ages of husbands and wives in the U.S. is"

    (a) exactly -1 (b) close to -1 (c) close to 0 (d) close to +1 (e) exactly +10
  2. Explain this statement: "Study time and test grade were perfectly positively correlated."
  3. Explain this statement: "The president is reported to have a 39% approval rating with a margin of error of 5%."
  4. If you could choose any level of significance to use in hypothesis testing, what would you choose and why?
  5. Give 2 ways to make a confidence interval smaller.

14. Consider the five income groups listed in the chart here.

Group% Distribution for the General Population
Under $15,000 23%
$15,000 - $34,999 33%
$35,000 - $49,999 17%
$50,000 - $74,000 16%
Over $75,000 11%

The data is from the Statistical Abstract of the U.S., 1993. 1000 black families were surveyed and their income followed this pattern:

  1. At the 1% significance level can we conclude that the distribution of income levels for blacks is different than that for the general population?
  2. Write your conclusion using P-value.

16. A study dealing with divorced couples gathered data on the length of time from marriage to separation. A random sample of 100 divorced couples had an average length of marriage of 5.9 years with a sample standard deviation of 2.0 years.

  1. Construct a 99% confidence interval for the mean length of time from marriage to separation for the population of all divorced couples.
  2. What sample size would be needed if we want to be 99% sure that the sample mean is within 0.25 year of the true population mean?

17. An automobile manufacturer is testing a new bumper that he feels will reduce the cost of repairs in front-end collisions at low speeds. Experience with the presently used bumper indicates that at 10 mph the cost of repair resulting from a front end collision has a mean of $325. To perform the test, the manufacturer equips 15 of his cars with the new bumper and has these 15 cars undergo front end collisions at 10 mph. Following this, the cars are repaired, and the cost of repair for each car is recorded. The sample mean x = $291.70 and the sample standard deviation s = $19.82.

  1. Perform the hypothesis test at the 5% level of significance.
  2. What is the name of the distribution that is used for this test?
  3. What criteria must be met to use this distribution?
  4. If the sample size was increased to over 30, how would this problem change? Would there be any advantage to having more than 30 in the sample?

18. A university science department has agreed upon Top 10% the grading policy shown here for an introductory course. The chairman of the department is not sure the policy was followed last semester. Since the introductory course has 2000 students in 50 sections, he does not want to tally all of these grades. Instead, he plans to take a random sample of 50 grades and perform a hypothesis test to see if the policy is followed.

The results of his sample are: A -- 10; B -- 10; C -- 10; D -- 14; F -- 6.

  1. Test the hypothesis at the 1% level of significance.
  2. Will your conclusion change if you change your level of significance to 5% ?

19. The following data gives the mean percent of items answered correctly on international mathematics tests in Algebra and Statistics for 8th grade students in the countries listed in 1981-1982. (Youth Indicators, 1988).

Country Algebra Score Statistics Score
England 40 60
Finland 44 58
France 55 57
Hong Kong 43 56
Hungary 50 60
Israel 44 52
Japan 60 71
Luxembourg 31 37
Netherlands 51 66
New Zealand 39 57
Nigeria 32 37
Scotland 43 59
Swaiziland 25 36
Sweden 32 56
Thailand 38 45
United States 42 58
  1. Draw a scatterdiagram for this data. Label clearly and COMMENT.
  2. List the correlation coefficient.
  3. Test the correlation coefficient, r, for significance at the 0.05 level of significance.
  4. If it is appropriate, list the equation of the regression line.
  5. Explain clearly and in sentence form the meaning of the slope of this line.
  6. Explain clearly and in sentence form the meaning of the y-intercept of this line.
  7. Give two points on the regression line. Use them to plot the regression line on the scatterdiagram.
  8. What is the best predicted mean percent of items answered correctly for statistics, if the algebra score is 45?

20. In a study of 200 accidents that required treatment in an emergency room, 80 of the accidents occurred at home.

  1. Find a 95% confidence interval for the true percentage of accidents treated in the emergency room that occur at home.
  2. Find a 99% confidence interval for the same value.
  3. Find a 90% confidence interval for the same value.
  4. Explain clearly the relationship between confidence level and the width of the confidence interval.

21. In 1985, 27% of Americans believed that "We should keep our doors open to immigration." In a survey of 1,108 adult Americans who participated in a Time Magazine/CNN survey in September, 1993, 24% then believed this statement.

  1. At the 5% level of significance, test the claim that the percentage of Americans who believe "We should keep our doors open to immigration" has fallen from 1985 to 1993.
  2. Would your conclusion change if you decreased σ to 1% ? EXPLAIN.

22. It is claimed that a new treatment for prolonging the lives of cancer patients is more effective than the standard one. With the standard treatment the mean survival period was shown to be 4.3 years. The new treatment was administered to 20 patients and the duration of their survival (measured in years) is recorded here:

3.2 5.6 4.2 6.2 1.8 9.4 6.2 8.1 2.5 5.1
2.4 3.8 7.1 4.0 1.7 0.6 9.2 7.7 5.1 6.9
  1. Is the claim of an increase in life span resulting from the new methods supported by the sample? Test at the 1% level of significance and make any appropriate assumptions.
  2. What was the name of the distribution that is used to solve this problem? What criteria must be met to use this distribution?

23. What role does the Central Limit Theorem play in inferential statistics?

24. The college president asks the statistics teachers to estimate the average age of the students at their college. How large a sample is necessary? The statistics teachers decide the estimate should be accurate to within 1 year and be at a 99% confidence level. From a previous study, the standard deviation of the ages was found to be 2.5 years.

25. The sociology department is conducting a study to see if the percentage of dual income families has increased over the past 15 years. Fifteen years ago, studies showed that in 45% of the families studied, both spouses worked outside the home. A study this year shows that for a random sample of 400 families in which both spouses were present, both spouses earn income in 210 of the cases.

  1. Test the hypothesis that the proportion of families with dual incomes has increased over the past 15 years. Use a 1% level of significance.
  2. What is the minimum level of significance at which we can reject H0 ?
  3. Find a 90% confidence interval for the proportion of families with dual incomes.

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