Deck 14: Tests of Hypotheses Based on Count Data

Question 1

It is claimed that at most $70 \%$ of all local businesses last more than one year. The null hypothesis $p=0.70$ is tested against the alternative $p>0.70$ at $\alpha=0.05$ for a sample size of $n=11$ . Suppose eight of the businesses lasted more than one year. Which of the following decisions should be made regarding the null hypothesis?

A) Reject, since $11>8$ .
B) Do not reject, since $11>8$ .
C) Reject, since $8>4$ .
D) Do not reject, since $8>4$ .

Answer 1

Do not reject, since $11>8$ .

Question 2

It is claimed that $40 \%$ of all shoppers at a shopping mall enter a particular department store. A random sample of 14 shoppers at the mall was selected. If the null hypothesis $p=0.40$ is tested against the alternative that $p<0.40$ , the null hypothesis will be rejected at $\alpha=0.01$ if __________ of the 14 shoppers enter the store.

A) at least 2
B) at most 1
C) at least 11
D) at most 10

Answer 2

at most 1

Question 3

The null hypothesis that two processes produce the same proportion of defectives can be written

A) $p=0$ .
B) $\frac{x_{1}}{n_{1}}-\frac{x_{2}}{n_{2}}=0$ .
C) $p_{1}-p_{2}=0$ .
D) $\frac{x}{n}=0$ .

Answer 3

$p_{1}-p_{2}=0$ .

Question 4

A placement office at a large university claims that approximately $70 \%$ of the school's graduates will obtain jobs in their major field upon graduation. The administration feels that the percentage is larger than $70 \%$ . From a random sample of 100 recent graduates it was found that 75 had obtained jobs in their field. The hypotheses are

A) $H_{0}: p_{1}-p_{2}=0.70$
$H_{\mathrm{A}}: p_{1}-p_{2}>0.70$
B) $H_{0}: p=0.70$
$H_{\mathrm{A}}: p>0.70$
C) $H_{0}: p<0.70$
$H_{\mathrm{A}}: p \geq 0.70$
D) $H_{0}: p=0.70$
$H_{\mathrm{A}}: p \neq 0.70$

Question 5

If the observed frequencies are exactly equal to the expected frequencies in a chi-square test, the value of $X^{2}$ is

A) close to 1 .
B) a large positive value.
C) 0 .
D) a very small positive value.

Question 6

A small computed $X^{2}$ value for a contingency table reveals which of the following concerning the two variables?

A) There is a strong relationship between them.
B) One is dependent on the other.
C) The observed frequencies greatly differ from the expected frequencies.
D) One is independent of the other.

Question 7

In a study to determine if a college student's year in college is independent of his or her major, a sample of 200 students was found to contain 40 juniors and 50 accounting majors. Then, the expected frequency of junior accounting majors is

A) 10 .
B) 20 .
C) 25 .
D) 15 .

Question 8

Which of the following conclusions is reasonable for a test of independence at $\alpha=0.05$ if $\mathrm{X}_{0.05}^{2}=18.31$ ?

A) If $X^{2}=17.50$ , then dependent.
B) If $X^{2}=19.15$ , then independent.
C) If $X^{2}=18.95$ , then dependent.
D) If $X^{2}=18.60$ , then independent.

Question 9

For a contingency table, which of the following is true?

A) Numbers of rows and columns are equal.
B) Each cell has frequency $\geq 5$ .
C) Column and row totals are left to chance.
D) It cannot be used to test for the equality of two populations.

Question 10

An office worker claims that at most $30 \%$ of the phone calls he makes are the nonbusiness type. Test the null hypothesis that $p=0.30$ against a one-tailed alternative if a random sample of 13 of his calls reveals 8 nonbusiness calls. Use $\alpha=0.05$ .

Question 11

An office worker claims that at most $30 \%$ of the phone calls he makes are the nonbusiness type. Test the null hypothesis that $p=0.30$ against a one-tailed alternative if a random sample of 13 of his calls reveals 8 nonbusiness calls. Use $\alpha=0.01$ .

Question 12

An office worker claims that at most $30 \%$ of the phone calls he makes are the nonbusiness type. Conduct a two-tailed test of $p=0.30$ at $\alpha=0.05$ , if a random sample of 13 of his calls reveals 8 nonbusiness calls.

Question 13

A cereal company is marketing a new breakfast food. In a random sample of 250 people who have tried the product, 175 said that they would buy it again. What can we state with $99 \%$ confidence about the maximum
size of our error if we use the sample proportion $\frac{175}{250}=0.70$ as an estimate of the actual population proportion of people who will buy the product a second time?

Question 14

A drug company wants to know whether the probability is really 0.20 that one of their products will produce side effects. In a random sample of 150 consumers of the product, 42 were victims of side effects. Conduct the two-tailed test. Use $\alpha=0.01$ . State the hypothesis.

Question 15

A drug company wants to know whether the probability is really 0.20 that one of their products will produce side effects. In a random sample of 150 consumers of the product, 42 were victims of side effects. Conduct a one-tailed test where the alternative is that the probability is greater than 0.20 . Use $\alpha=0.01$ .

Question 16

A suntan lotion manufacturer wants to estimate the proportion of people who will get a tan by using their product. In a sample of 80 people, 50 were tanned by the product. Find a $98 \%$ confidence interval for the population proportion of people who will receive a tan from the lotion.

Question 17

A university placement office wants to estimate the percentage of graduates that obtain jobs within one month of graduation. They want to be $98 \%$ confident in their result and be within $5 \%$ of the true percentage. Find the necessary sample size for this experiment, and find a $98 \%$ confidence interval if $30 \%$ of the sample received jobs within one month of graduation.

Question 18

A television station wants to determine if there is a difference in the proportions of people who watched two of their programs. In random samples of 60 and 80 people, 25 and 40 people watched the first and second programs respectively.

-Using a 5\% significance level, conduct the test for the situation above.

Question 19

A television station wants to determine if there is a difference in the proportions of people who watched two of their programs. In random samples of 60 and 80 people, 25 and 40 people watched the first and second programs respectively.

-In the situation above, suppose the station wants to know if fewer people watched the first program than the second. Use the same sample results and conduct the one-tailed test at $\alpha=0.05$ .

Question 20

A manufacturer has test-marketed a new men's cologne. Simple random samples of male students at a university were asked to try the product for four weeks and then were asked the question: "Would you purchase the product?" The results are shown in the following table:


-Would you conclude the respondents' preferences are independent of year in college? Conduct the test at $\alpha=0.05$ .

Question 21

A sample of 210 people were telephoned. The number of rings that have fully elapsed before each person answered the phone are recorded below.


-At the 0.05 level of significance, does it appear that the data may be looked upon as a random sample from a binomial population with $p=0.4$ and $n=6$ ? Conduct the goodness-of-fit test.

Question 22

A company is marketing a laser disk recorder. A survey of 80 randomly selected male students at a college reveals that 45 would be interested in buying such a recorder. A corresponding survey of 120 female students reveals that 75 want to buy it. Test the hypothesis that there is no difference between the males and the females on this issue. Use the $5 \%$ significance level.

Question 23

Ninety out of 120 house-husbands prefer detergent $A$ . If the 90 house-husbands represent a random sample from a population of all potential purchasers, estimate the fraction of total house-husbands favoring detergent $A$ by constructing a $98 \%$ confidence interval.

Question 24

An auditor is assigned to investigate the probability of a bank's accounts having errors. If she has reason to believe that the probability is anywhere between 0.15 and 0.40 , how large a sample will she need to be $99 \%$ confident that the estimated percentage of accounts containing errors is within 0.02 of the true percentage?

Question 25

An automobile repair service asks 170 customers with annual incomes under \$20,000, 180 customers with annual incomes from $\$ 20,000$ to $\$ 50,000$ and 150 customers with annual incomes over $\$ 50,000$ whether they rated the repair service as outstanding, above average, average, below average, or poor. What hypotheses do we want to test if we are going to perform a chi-square analysis of the resulting $5 \times 3$ table?

Question 26

In determining the necessary sample size for $95 \%$ confidence interval, the value 0.95 must be substituted into the formula for $z_{\alpha / 2}$ .

Question 27

The chi-square test can be used to determine if there is a significant difference between two sample proportions.

Question 28

In order to use the maximum error of estimate formula involving the sample value $x / n, n$ must be large enough to justify the normal curve approximation to the binomial distribution.

Question 29

To decide whether or not observed differences among three sample proportions can be attributed to chance requires the use of the normal distribution table.

Question 30

The value of $\alpha$ has a definite effect on the number of degrees of freedom used to determine $\mathrm{X}^{2}$ .

Question 31

For a contingency table, the expected frequency values are based on the assumption that the null hypothesis is true.

Question 32

The null hypothesis in a chi-square test of independence is that there is no relationship between the two variables.

Question 33

The $X^{2}$ statistic should never be applied to contingency tables if some of the expected frequencies are less than 5.

Question 34

The null hypothesis in a goodness-of-fit test is that the distribution of the sample does not fit the theoretical distribution.

Question 35

The symbol $x / n$ represents a sample estimate of __________.

Question 36

The standard error of the proportion is given by __________.

Question 37

If $n=400$ , an hypothesis test may be conducted to test a population proportion using the normal curve if $p$ is between __________ and __________.

Question 38

For a particular error value and confidence level, the maximum value that the sample size formula can assume occurs when $p$ equals __________.

Question 39

The chi-square statistic for a $7 \times 4$ contingency table will have __________ degrees of freedom.

Question 40

The sample value $\frac{x_{1}}{n_{1}}-\frac{x_{2}}{n_{2}}$ is an estimate of the population value __________.

Question 41

When conducting a chi-square test for a goodness-of-fit, the number of degrees of freedom is given by __________.

Question 42

To determine if a set of data fits the pattern of the binomial distribution, the __________ test is used.

Question 43

The larger the statistic in a goodness-of-fit test, the __________ likely the data fits the hypothesized distribution.