The following questions will be based on the data given by the class in the beginning of the quarter.

1) The profile of the University given on the university web

page (www.washington.edu) states that about 53% (0.53) of

the University undergraduate population is women. Using

our class as a sample, let’s test that claim at alpha=0.05.

Men: 14, Women: 21

a) Write the null and alternative hypotheses.

b) Decide on the sampling distribution and draw a picture

of the rejection region.

c) Calculate your test statistic.

d) Make a decision.

e) Give some reasons why your decision might have occurred,

as well as potential causes for error in your decision.

f) Repeat the test if the University had claimed that there

are no more than 53% women at the university.

A) H₀: p=0.53; H_a: p≠0.53

B) np=18.55, nq=16.45, can use normal distribution, with z_c = +/- 1.96

C)

D) fail to reject H₀
2) The “Common data set” given on the university web page

states that the average age of all undergraduate students

at the University of Washington is 21.0 years old. The

average age of 20 randomly selected students in our class

is 22.4 with a standard deviation of 3.80. Do we have enough evidence to reject the claim that the average age is 21.0? Assume age is normally distributed.

a) Write the null and alternative hypotheses.

b) Decide on the sampling distribution and your level of

significance, then draw a picture of the rejection

region.

c) Calculate the test statistic.

d) Make a decision.

e) Give some reasons why your decision might have occurred,

as well as potential causes for error in your decision.

A) H₀: μ=21; H_a: μ≠21

B) Since σ is unknown and n<30, use t distribution with 19 df, t_9,0.05 = +/-2.093, t_9,0.10 = +/- 1.729

C)

D) Fail to reject H₀
3) Upon taking a closer look at the web page, the university

reports that the average age of students is 21.0 with a

standard deviation of 2.36.

Does this change our test? If so, repeat the test and

discuss the results.

Now we can use a normal distribution because the population standard deviation is known:

z_0.05 = +/-1.96

Remember to use the population standard deviation to calculate the test statistic!

In this case we can reject H₀ at α=0.05. When we have more information we have more power to reject the null hypothesis.

4) I claim that the standard deviation of heights in the

entire Center for Quantitative Science is 5.12 or more. The

standard deviation of heights for 35 people in our class is

4.54. Assuming the heights are normally distributed, is

there enough evidence to reject the claim?

a) Write the null and alternative hypotheses.

b) Decide on the sampling distribution and your level of

significance, then draw a picture of the rejection

region.

c) Calculate the test statistic.

d) Make a decision.

e) Give some reasons why your decision might have occurred,

as well as potential causes for error in your decision.

Note: For this problem I told the class to use 30 df, because 34 is not included on the table

A) H₀: σ≥5.12; H_a: σ<5.12

B) use χ² distribution: with α=0.05, χ²_l = 18.493 (left-tailed test), reject if statistic is less than 18.493

C)

D) Fail to reject H₀. Just to compare, if I had used 40 df (the next line in the table), the critical value would have been 26.509, in which case I still would have failed to reject. That tells us that our true critical value for 34 df is somewhere between 18.493 and 26.509, and our test statistic is not less than any number in that range. We are safe in rejecting H₀ even though we don’t have an exact critical value.