10.1

Two Population Means with Unknown Standard Deviations

Use the following information to answer the next 15 exercises:

Indicate if the hypothesis test is for

independent group means, population standard deviations, and/or variances known

independent group means, population standard deviations, and/or variances unknown

matched or paired samples

single mean

two proportions

single proportion

1

.

It is believed that 70% of males pass their drivers test in the first attempt, while 65% of females pass the test in the first attempt. Of interest is whether the proportions are in fact equal.

2

.

A new laundry detergent is tested on consumers. Of interest is the proportion of consumers who prefer the new brand over the leading competitor. A study is done to test this.

3

.

A new windshield treatment claims to repel water more effectively. Ten windshields are tested by simulating rain without the new treatment. The same windshields are then treated, and the experiment is run again. A hypothesis test is conducted.

4

.

The known standard deviation in salary for all mid-level professionals in the financial industry is $11,000. Company A and Company B are in the financial industry. Suppose samples are taken of mid-level professionals from Company A and from Company B. The sample mean salary for mid-level professionals in Company A is $80,000. The sample mean salary for mid-level professionals in Company B is $96,000. Company A and Company B management want to know if their mid-level professionals are paid differently, on average.

5

.

The average worker in Germany gets eight weeks of paid vacation.

6

.

According to a television commercial, 80% of dentists agree that Ultrafresh toothpaste is the best on the market.

7

.

It is believed that the average grade on an English essay in a particular school system for females is higher than for males. A random sample of 31 females had a mean score of 82 with a standard deviation of three, and a random sample of 25 males had a mean score of 76 with a standard deviation of four.

8

.

The league mean batting average is 0.280 with a known standard deviation of 0.06. The Rattlers and the Vikings belong to the league. The mean batting average for a sample of eight Rattlers is 0.210, and the mean batting average for a sample of eight Vikings is 0.260. There are 24 players on the Rattlers and 19 players on the Vikings. Are the batting averages of the Rattlers and Vikings statistically different?

9

.

In a random sample of 100 forests in the United States, 56 were coniferous or contained conifers. In a random sample of 80 forests in Mexico, 40 were coniferous or contained conifers. Is the proportion of conifers in the United States statistically more than the proportion of conifers in Mexico?

10

.

A new medicine is said to help improve sleep. Eight subjects are picked at random and given the medicine. The means hours slept for each person were recorded before starting the medication and after.

11

.

It is thought that teenagers sleep more than adults on average. A study is done to verify this. A sample of 16 teenagers has a mean of 8.9 hours slept and a standard deviation of 1.2. A sample of 12 adults has a mean of 6.9 hours slept and a standard deviation of 0.6.

12

.

Varsity athletes practice five times a week, on average.

13

.

A sample of 12 in-state graduate school programs at school A has a mean tuition of $64,000 with a standard deviation of $8,000. At school B, a sample of 16 in-state graduate programs has a mean of $80,000 with a standard deviation of $6,000. On average, are the mean tuitions different?

14

.

A new WiFi range booster is being offered to consumers. A researcher tests the native range of 12 different routers under the same conditions. The ranges are recorded. Then the researcher uses the new WiFi range booster and records the new ranges. Does the new WiFi range booster do a better job?

15

.

A high school principal claims that 30% of student athletes drive themselves to school, while 4% of non-athletes drive themselves to school. In a sample of 20 student athletes, 45% drive themselves to school. In a sample of 35 non-athlete students, 6% drive themselves to school. Is the percent of student athletes who drive themselves to school more than the percent of nonathletes?

Use the following information to answer the next five exercises:

A teacher predicts that the distribution of grades on the final exam will be and they are recorded in

Table 11.27

.

Grade

Proportion

A

0.25

B

0.30

C

0.35

D

0.10

Table

11.27

The actual distribution for a class of 20 is in

Table 11.28

.

Grade

Frequency

A

7

B

7

C

5

D

1

Table

11.28

9

.

d

f

=

d

f

=

______

10

.

State the null and alternative hypotheses.

11

.

Ï‡2

test statistic = ______

12

.

p

-value = ______

13

.

At the 5% significance level, what can you conclude?

Use the following information to answer the next nine exercises:

The following data are real. The cumulative number of AIDS cases reported for Santa Clara County is broken down by ethnicity as in

Table 11.29

.

Ethnicity

Number of Cases

White

2,229

Hispanic

1,157

Black/African-American

457

Asian, Pacific Islander

232

Total = 4,075

Table

11.29

The percentage of each ethnic group in Santa Clara County is as in

Table 11.30

.

Ethnicity

Percentage of total county population

Number expected (round to two decimal places)

White

42.9%

1748.18

Hispanic

26.7%

Black/African-American

2.6%

Asian, Pacific Islander

27.8%

Total = 100%

Table

11.30

14

.

If the ethnicities of AIDS victims followed the ethnicities of the total county population, fill in the expected number of cases per ethnic group.

Perform a goodness-of-fit test to determine whether the occurrence of AIDS cases follows the ethnicities of the general population of Santa Clara County.

15

.

H0

: _______

16

.

Ha

: _______

17

.

Is this a right-tailed, left-tailed, or two-tailed test?

18

.

degrees of freedom = _______

19

.

Ï‡2

test statistic = _______

20

.

p

-value = _______