stats 3-21

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a)A worldwide organization of academics claims that the mean IQ score of its members is 118, with a standard deviation of 17. A randomly selected group of 35 members of this organization is tested, and the results reveal that the mean IQ score in this sample is 114.2. If the organization's claim is correct, what is the probability of having a sample mean of 114.2 or less for a random sample of this size?

b) Suppose that 57% of all college seniors have a job prior to graduation. If a random sample of 50 college seniors is taken, approximate the probability that at most 31 have a job prior to graduation. Use the normal approximation to the binomial with a correction for continuity.

Round your answer to at least three decimal places. Do not round any intermediate steps.

c) A worldwide organization of academics claims that the mean IQ score of its members is 118, with a standard deviation of 17. A randomly selected group of 35 members of this organization is tested, and the results reveal that the mean IQ score in this sample is 114.2. If the organization's claim is correct, what is the probability of having a sample mean of 114.2 or less for a random sample of this size?

Carry your intermediate computations to at least four decimal places. Round your answer to at least three decimal places.

d) Suppose that the lifetimes of tires of a certain brand are normally distributed with a mean of 74,000 miles and a standard deviation of σ miles. These tires come with a 60,000-mile warranty. The manufacturer of the tires can adjust σ during the production process, but the adjustment of σ is quite costly. The manufacturer wants to set σ once and for all so that only 1% of the tires will fail before warranty expires. Find the standard deviation to be set. Carry your intermediate computations to at least four decimal places. Round your answer to at least one decimal place.

e) Let Z be a standard normal random variable. Use the calculator provided, or this table, to determine the value of c.

=P( ≤c≤Z− 0.54)=0.2571

Carry your intermediate computations to at least four decimal places. Round your answer to two decimal places.

14) A recent study at a local college claimed that the proportion, p, of students who commute more than fifteen miles to school is no more than 20%. If a random sample of 265 students at this college is selected, and it is found that 71 commute more than fifteen miles to school, can we reject the college's claim at the level of 0.01 significance?

Perform a one-tailed test. Then complete the parts below.

Carry your intermediate computations to three or more decimal places and round your answers as specified in the table. (If necessary, consult a list of formulas.)

(a) State the null hypothesis and the alternative hypothesis .

(b) Determine the type of test statistic to use.

(d) Find the p-value. (Round to three or more decimal places.)

(e) Can we reject the claim that the proportion of students who commute more than fifteen miles to school is no more than 20%?

Yes No

15) A psychologist specializing in marriage counseling claims that, among all married couples, the proportion p for whom her communication program can prevent divorce is at least 78% . In a random sample of 245 married couples who completed her program, 177 of them stayed together. Based on this sample, can we reject the psychologist's claim at the 0.1 level of significance?

Perform a one-tailed test. Then complete the parts below.

Carry your intermediate computations to three or more decimal places and round your answers as specified in the table. (If necessary, consult a list of formulas.)

(a) State the null hypothesis and the alternative hypothesis .

(b) Determine the type of test statistic to use.

(d) Find the critical value. (Round to three or more decimal places.)

(e) Can we reject the psychologist's claim that the proportion of married couples for whom her program can prevent divorce is at least 78%?

Yes No

16) A hospital claims that the proportion, p, of full-term babies born in their hospital that weigh more than 7 pounds is 47%. In a random sample of 160 babies born in this hospital, 58 weighed over 7 pounds. Is there enough evidence to reject the hospital's claim at the 0.05 level of significance?

Perform a two-tailed test. Then complete the parts below.

Carry your intermediate computations to three or more decimal places and round your answers as specified in the table. (If necessary, consult a list of formulas.)

(a) State the null hypothesis and the alternative hypothesis .

(b) Determine the type of test statistic to use.

(d) Find the p-value. (Round to three or more decimal places.)

(e) Can we reject the claim that the proportion of full-term babies born in their hospital that weigh more than 47% pounds is?

Yes No

17) A recent study at a local college claimed that the proportion, p , of students who commute more than fifteen miles to school is no more than 25% . If a random sample of 275 students at this college is selected, and it is found that 76 commute more than fifteen miles to school, can we reject the college's claim at the 0.05 level of significance?

Perform a one-tailed test. Then complete the parts below.

Carry your intermediate computations to three or more decimal places and round your answers as specified in the table. (If necessary, consult a list of formulas.)

(a) State the null hypothesis and the alternative hypothesis .

(b) Determine the type of test statistic to use.

(d) Find the critical value. (Round to three or more decimal places.)

(e) Can we reject the claim that the proportion of students who commute more than fifteen miles to school is no more than 25%?

Yes No

18) The mean SAT score in mathematics, u, is 553. The standard deviation of these scores is 38. A special preparation course claims that its graduates will score higher, on average, than the mean score 553. A random sample of 35 students completed the course, and their mean SAT score in mathematics was 565. Assume that the population is normally distributed. At the 0.01 level of significance, can we conclude that the preparation course does what it claims? Assume that the standard deviation of the scores of course graduates is also 38.

Perform a one-tailed test. Then complete the parts below.

Carry your intermediate computations to three or more decimal places, and round your responses as specified below. (If necessary, consult a list of formulas.)

(a) State the null hypothesis and the alternative hypothesis .

(b) Determine the type of test statistic to use.

(d) Find the critical value. (Round to three or more decimal places.)

(e) Can we support the preparation course's claim that its graduates score higher in SAT?

Yes No

19) It seems these days that college graduates who are employed full-time work more than 40 -hour weeks. Data are available that can help us decide if this is true. A survey was recently sent to a group of adults selected at random. There were 13 respondents who were college graduates employed full-time. The mean number of hours worked per week by these 13 respondents was 42 hours, with a standard deviation of 10 hours.

Assume that the population of hours worked per week by college graduates employed full-time is normally distributed with mean u. Can we conclude that u is greater than 40 hours? Use the 0.1 level of significance.

Perform a one-tailed test. Then complete the parts below.

Carry your intermediate computations to three or more decimal places and round your answers as specified in the table. (If necessary, consult a list of formulas.)

(a) State the null hypothesis and the alternative hypothesis .

(b) Determine the type of test statistic to use.

(d) Find the p-value. (Round to three or more decimal places.)

(e) Can we conclude, at the 0.1 level of significance, that the mean number of hours worked per week by college graduates is greater than 40 hours?

Yes No

20) A psychologist specializing in marriage counseling claims that, among all married couples, the proportion p for whom her communication program can prevent divorce is at least 77%. In a random sample of 230 married couples who completed her program, 160 of them stayed together. Based on this sample, can we reject the psychologist's claim at the 0.01 level of significance?

Perform a one-tailed test. Then complete the parts below.

Carry your intermediate computations to three or more decimal places and round your answers as specified in the table. (If necessary, consult a list of formulas.)

(a) State the null hypothesis and the alternative hypothesis .

(b) Determine the type of test statistic to use.

(d) Find the p-value. (Round to three or more decimal places.)

(e) Can we reject the psychologist's claim that the proportion of married couples for whom her program can prevent divorce is at least 077% ?

Yes No