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KEY AP Stats Review Chapter 13 and 14
- A simple random sample of 100 high school seniors was selected from a large school district. The gender of each student was recorded, and each student was asked the following questions. I. Have you ever had a part-time job? II. If you answered yes to the previous question, was your part-time job in the summer only?
The responses are summarized in the table below.
Gender Job Experience Male Female Total Never had a part-time job 21 31 52 Had a part-time job during summer only 15 13 28 Had a part-time job but not only during summer 12 8 20 Total 48 52 100
a. If a high school senior is to be selected at random from this sample, what is the probability that the selected senior is male or had a part-time job only during the summer? 61/ b. If a high school senior who is a female is to be selected at random from this sample, what is the probability that the selected senior never had a part-time job? 31/ c. Which test of significance should be used to test if there is an association between gender and job experience for the population of high school seniors in this district? State the null and alternative hypotheses for the test. Use the Chi-Square Test of Independence. H 0 : There is no association between Gender and Job Experience. Ha: There is an association between Gender and Job Experience. d. Carry out the test identified in part (c) at the 5% significance level. Conditions: (1) SRS-stated in the problem. (2) All expected counts are at least 5. Yes, based on the Expected values
shown here [ ].
Test statistic: ∑ (^ )^ with d.f. = (3 – 1)(2 – 1) = 2 P-value: 0. Since 0.258 > 0.05 we fail to reject the null hypothesis. Based on our sample, there is insufficient evidence to conclude that any association between gender and job experience exists.
- For many years the medically accepted practice of giving aid to a person experiencing a heart attack was to have the person who placed the emergency call administer chest compression (CC) plus standard mouth-to-mouth resuscitation (MMR) to the heart attack patient until the emergency response arrived. However, some researchers believe that CC alone would be a more effective approach.
In the 1990s a study was conducted in Seattle in which 518 cases were randomly assigned to treatments: 278 to CC plus standard MMR and 240 to CC alone. A total of 64 patients survived the heart attack: 29 in the group receiving CC plus standard MMR, and 35 in the group receiving CC alone. a. Complete a test of significance at the α = 0.05 level based on the following hypotheses. H 0 : The survival rates for the two treatments are equal. Ha: The treatment that uses CC alone produces a higher survival rate.
Parameter: We are interested in comparing the difference in population proportions of individuals receiving CC plus standard MMR and those receiving CC alone.
Hypotheses: H 0 :.
Ha:
Assumptions: SRS-states random assignment, Normality-more than 30 in each group so CLT applies, Independence- each patient should be independent from the next.
Name of Test: One-tailed two-proportion z-test
Test Statistic: ̂ ̂ √ ̂ (̂ )( )
where ̂= 0.1043, ̂= 0.1458, ̂= 0.1236, ,
Obtain p-value: p -value = 0.0761 which means we could expect this result to occur by chance nearly 8 out of 100 times in repeated samples.
Make your decision: Fail to reject H 0 at the 5% level
Conclusion: We have insufficient evidence to support the claim that CC alone produces a higher survival rate.
b. Based on the results of the test and the fact that “some researchers believe that CC alone would be a more effective approach,” which type of error (Type I or Type II) could have been made? What is one potential consequence of this error? A Type I error is being committed since the null hypothesis is “true” but is being rejected. One potential consequence is a greater loss of life.
- A large pet store buys the identical species of adult tropical fish from two different suppliers—Buy-Rite Pets and Fish Friends. Several of the managers at the pet store suspect that the lengths of the fish from Fish Friends are consistently greater than the lengths of the fish from Buy-Rite Pets. Random samples of 8 adult fish of the species from Buy-Rite Pets and 10 adult fish of the same species from Fish Friends were selected and the lengths of the fish, in inches, were recorded, as shown in the table below.
Length of Fish Mean Standard Deviation Buy-Rite Pets 3.4, 2.7, 3.3, 4.1, 3.5, 3.4, 3.0, 3.8 3.40 0. Fish Friends 3.3, 2.9, 4.2, 3.1, 4.2, 4.0, 3.4, 3.2, 3.7, 2.6 3.46 0.
Do the data provide convincing evidence that the mean length of the adult fish of the species from Fish Friends is greater than the mean length of the adult fish of the same species from Buy-Rite Pets?
Parameter: We are interested in comparing the true mean length, μ 1 , of adult fish from Buy-Rite Pets with the true mean length, μ 2 , of adult fish from Fish Friends.
Hypotheses: H 0 : μ 1 = μ 2 Ha: μ 1 < μ 2
Assumptions: SRS – we must assume random selection, Normality – parallel boxplots show approximate normality Buy-Rite Pets
Fish Friends
Skunk 100
Test Statistic: ̅ ̅ √
c. State another set of hypotheses that might be used to test the researchers’ belief. Test these at the 5% significance level also.
Answers may vary as there are several possibilities.
- Contestants on a game show spin a wheel like the one shown. Each of the four outcomes on this wheel is equally likely and outcomes are independent form one spin to the next.
The contestant spins the wheel. If the result is a skunk, no money is won and the contestant’s turn in finished. If the result is a number, the corresponding amount in dollars is won. The contestant can then stop with those winnings or can choose to spin again, and his or her turn continues. If the contestant spins again and the result is a skunk, all of the money earned on that turn is lost and the turn ends. The contestant may continue adding to his or her winnings until he or she chooses to stop or until a spin results in a skunk. a. What is the probability that the result will be a number on all of the first three spins on the wheel?
( )
b. A contestant who lost this game alleges that the wheel is not fair. In order to check on the fairness of the wheel, the data in the table below were collected for 100 spins of this wheel.
Result Skunk $100 $200 $ Frequency 33 21 20 26
Based on these data, can you conclude that the four outcomes on this wheel are not equally likely? Give appropriate statistical evidence to support your answer.
Hypotheses:
Ho: The four outcomes are equally likely.
Ha: The four outcomes are not equally likely.
Name of Test: Chi-Square Goodness of Fit test
Conditions: SRS – outcomes are random, All expected values are at least 5
Test Statistic: ∑^ ( )
Fail to reject H 0 since 0.2367 > .05. Thus, we can conclude that there is insufficient evidence at the 5% level to support the claim that the game is unfair.
