Stats Online Ch. 15

In a rural area only about 18 % of the wells that are drilled find adequate water at a depth of 100 feet or less. A local man claims to be able to find water by​ dowsing, that​ is, using a forked stick to indicate where the well should be drilled. You check with 66 of his customers and find that 16 have wells less than 100 feet deep. Answer the questions below. Assume the independence assumption is met.

(a) Write appropriate hypotheses. H o : The percentage of successful wells drilled by the dowser """is equal to 18%.""" Ha : The percentage of successful wells drilled by the dowser """is greater than 18%.""" ​(b) Check the necessary assumptions. The independence assumption is """satisfied.""" The randomization condition is """not satisfied.""" The​ 10% condition is """satisfied.""" The​ success/failure condition is """satisfied.""" (c) Perform the mechanics of the test. What is the​ P-value? The sample size is n=66 the sample proportion Po=.18 The sample proportion is p^=16/66=0.242 The value of qo is 1-.18=.82 0.242-.18/square root (.18)(.82)/66=1.31 ​(d) Explain carefully what the​ P-value means in this context. If his dowsing has the same success rate as standard drilling​ methods, the​ P-value is the probability of seeing results as good as those of the​ dowser, or​ better, because of natural sampling variation. (e) What is your​ conclusion? (Consider a​ P-value of around​ 5% to represent strong​ evidence.) We fail to reject the null hypothesis. There is not evidence to suggest that the dowser has a success rate higher than 18 %.

In 2002 a national vital statistics report indicated that about 3.3 % of all births produced twins. Is the rate of twin births the same among very young​ mothers? Data from a large city hospital found only 11 sets of twins were born to 495 teenage girls. Test an appropriate hypothesis and state your conclusion. Be sure the appropriate assumptions and conditions are satisfied before you proceed.

Are the assumptions and the conditions to perform a​ one-proportion z-test​ met? Yes State the null and alternative hypotheses. Choose the correct answer below. H0​: p = 0.033 HA​: p not= 0.033 nq0 = 495*.033 = 16 nq0 = 495*(1-.033) = 495 q0 = 1-.033=.967 p^= 11/495 = .022 square root (.033)(.967)/495 = .00803 z = .022-.033/.00803= -1.37 p-value = 2*.085=.170 Fail to reject Ho. The proportion of twin births for teenage mothers is not different from the proportion of twin births for all mothers.

Sam ran an experiment to test optimum power and time settings for microwave popcorn. His goal was to deliver popcorn with fewer than 8​% of the kernels left​ unpopped, on average. He determined that power 9 at 4 minutes was the best combination. To be sure that the method was​ successful, he popped 8 more bags of popcorn​ (selected at​ random) at this setting. All were of high​ quality, with the percentages of unpopped kernels shown below. 3.7​, 10.7​, 4.6​, 5.8​, 6.1​, 9.9​, 12.3​, 4.9

Ho: u = 8 Ha: u <8 Claimed Hypothesis Mean, H0: 8 Sample Mean, x: 7.25 Standard deviation, σ: 3.2293 Sample Size, n: 8 t= -0.657 p-value = 0.2661 Yes​, there is enough evidence suggest that less than 11​% of the kernels are left unpopped when the specified power and time settings are used.

A​ company's old decongestant formula provided relief for 64​% of the people who used it. The company tests a new formula to see if it is​ better, and gets a​ P-value of 0.215. Is it reasonable to conclude that the new formula and the old one are equally​ effective? Explain.

It is not reasonable to conclude that the new formula and the old one are equally effective. The​ P-value cannot suggest a conclusion.

Your friend in your statistics class is upset about a recent increase in the price to wash and dry a load of laundry. She wants to conduct a one proportion​ z-test to see if more than half the residents in her dorm oppose the increase. She will poll a random sample of 30 residents. Which Normal model will she​ use?

N(0.50,0.091)

One of your friends in your statistics class has polled 15 people in her dorm about a recent increase in the price to wash and dry a load of laundry. She wants to conduct a one proportion​ z-test to see if more than half the residents oppose the increase. Is a Normal model appropriate​ here?

No

The probability of seeing sample data like those seen​ (or something even less​ likely) given the null hypothesis is true is the​

P-value

A garden center wants to store leftover packets of vegetable seeds for sale the following​ spring, but the center is concerned that the seeds may not germinate at the same rate a year later. The manager finds a packet of last​ year's green bean seeds and plants them as a test. Although the packet claims a germination rate of 92​%, only 215 of 250 test seeds sprout. Is this evidence that the seeds have lost viability during a year in​ storage? Test an appropriate hypothesis and state your conclusion. Be sure the appropriate assumptions and conditions are satisfied before you proceed.

The independence assumption """is not""" justified. The​ 10% condition """is plausibly""" satisfied. The​ success/failure condition """is""" satisfied. Write appropriate hypotheses. H 0​: p = 0.92 and H A​: p not = 0.92 Determine the​ z-statistic. Select the correct choice below and fill in any answer boxes within your choice. z= -4.20 p-value = .000 State your conclusion assuming a​ P-value smaller than 0.05 is significant. This is strong evidence that less than 92​% of​ one-year-old seeds will germinate.

Which of the following is not a condition to check to estimate the mean of a​ population?

There are at least 10 successes and 10 failures in the sample.

Which of the following is a correct interpretation of a​ P-value that is not very​ small?

What we saw in the sample data is not​ surprising, so we fail to reject the null hypothesis.

Suppose the average 18-24-year old has 644 Facebook friends. A student wanted to test if the mean number is higher at his school. Using the data​ given, test an appropriate hypothesis and write a couple of sentences summarizing what you discover.

Write appropriate hypotheses for the test. H 0​: u = 644 H A​: u > 644 t= 5.14 p value = .000 There """is""" sufficient evidence to conclude that the mean number of Facebook friends for students at the school is """greater than""" 644.

Census data for a certain county show that 17​% of the adult residents are Hispanic. Suppose 78 people are called for jury duty and only 9 of them are Hispanic. Does this apparent underrepresentation of Hispanics call into question the fairness of the jury selection​ system? Explain.

Write appropriate hypotheses. H 0​: p = 17 % H A​: p < 17 % z = -1.28 p value = .100 """Do not reject""" the null hypothesis. There """is not""" sufficient evidence to conclude that the proportion of Hispanics in the jury pool is """less than""" the proportion of Hispanics in the population.

Suppose I want to test whether​ students' textbook costs for this semester are higher on average than the average cost of​ $450 in 2000. What would be an appropriate alternative​ hypothesis?

greater than 450

A student on the track team wants to test whether her times to run a mile are different after a new training regimen.​ Previously, her average time to run a mile was 8 minutes. Which is an appropriate alternative​ hypothesis?

not equals 8

Which of the following alternative hypotheses is​ two-sided (or​ two-tailed)?

p not equals 0.75

Which of the following hypotheses has the appropriate form for a null​ hypothesis?

p=0.30

Reject the null hypothesis if the​ p-value is​

small enough

Developing a new drug can be an expensive​ process, resulting in high costs to patients. A pharmaceutical company has developed a new drug to reduce​ cholesterol, and it will conduct a clinical trial to compare the effectiveness to the most widely used current treatment. The results will be analyzed using a hypothesis test.

​a) If the test yields a low​ P-value and the researcher rejects the null hypothesis that the new drug is not more​ effective, but it actually is not​ better, what are the consequences of such an​ error? People will pay more money for a drug that is no better than the most widely used current treatment. The company may lose some of its reputation for reliable quality and honest advertising. ​b) If the test yields a high​ P-value and the researcher fails to reject the null​ hypothesis, but the new drug is more​ effective, what are the consequences of such an​ error? Select all that apply. The company will lose potential revenue from selling the drug. People will miss an opportunity to use a more effective treatment for cholesterol.

According to a recent​ census, 18​% of the people in the United States are of Hispanic origin. One county supervisor believes her county has a different proportion of Hispanic people than the nation as a whole. She looks at their most recent survey​ data, which was a random sample of 436 county​ residents, and found that 49 of those surveyed are of Hispanic origin.

​a) State the hypotheses. Ho : p = 0.18 Ha : p not= 0.18 b) Name the model and check appropriate conditions for a hypothesis test. ​One-proportion z test 436* .18= 78.5 successes 436*.82= 357.5 failures ​Thus, the conditions are """met""" and it is """appropriate""" to use a Normal model and perform the hypothesis test. z= -3.67 p value = .000 d.) State the conclusion. Because the​ P-value is so​ low, there is evidence that the Hispanic population in this county differs from that of the nation as a whole.

A very large study showed that aspirin reduced the rate of first heart attacks by 51​%. A pharmaceutical company thinks it has a drug that will be more effective than​ aspirin, and plans to do a randomized clinical trial to test the new drug.

​a) The alternative hypothesis is more naturally """ one-sided """ because the company would like to determine if the new drug is """ more effective """ than aspirin. b) If the​ P-value is 0.0034​, then there """ is sufficient """ evidence to conclude that the new drug is more effective than aspirin because the​ P-value is """ small """. c) If the​ P-value is 0.34​, then there """ is not sufficient """ evidence to conclude that the new drug is more effective than aspirin because the​ P-value is """ large """.

A company claimed that every​ 18-ounce bag of their chocolate chip cookies contained at least 1000 chocolate chips. Dedicated statistics students at a school purchased some randomly selected bags of cookies and counted the chocolate chips. Some of their data are given below. 1224 1228 1096 1213 1409 1114 1336 1345 1232 1253 1345 1134 1187 1285 1285 1124

​a) The independence condition """is plausibly""" satisfied and the Nearly Normal Condition """is plausibly""" satisfied.​Therefore, using the​ Student-t model for inference """should be""" reliable. ​ b) Write appropriate hypotheses for the test. H 0​: u= 1000 H A​: u > 1000 The test statistic is t= 10.26 p value = .000 If the​ P-value is less than the chosen level of​ significance, then reject the null hypothesis. Otherwise do not reject the null hypothesis. Then interpret these results in the context of the given situation. Using a= 0.05 with a​ P-value of 0.000​, """reject""" the null hypothesis. """Reject""" the null hypothesis. There """is""" sufficient evidence to conclude that the mean number of chips is """greater than""" about 1000.