Q: A survey of 948 cell phone users found 472 have one number, 331 have two, 111 have three, and 34 have four or more. The guide claims the proportions are 52%, 34%, 8%, and 6%. What is the expected frequency for the 'three numbers' category?

$$E = 948 \times 0.08 = 75.84$$

Question 1

Which of the following is the correct formula for the test statistic in a two-sample hypothesis test for the difference of means when population variances are known?

Accepted Answer

$$z=\frac{x_1$-x_2$-D_0$}{\sqrt{\frac{\$$sigma_1$$^2}{n_1$}+\frac{\$$sigma_2$$^2}{n_2$}}}$$

Question 2

A researcher wants to test if the distribution of cell phone users by number of cell numbers matches the guide's reported percentages (52%, 34%, 8%, 6%). Which statistical test should be used?

Accepted Answer

Chi-square goodness-of-fit test

Question 3

In a multinomial (goodness-of-fit) test, what is the formula for the expected frequency $$E_i$$ for category $$i$$?

Accepted Answer

$$E_i = n p_{i0}$$

Question 4

A study of fatal DWI crashes over the days of the week is conducted to test if accidents occur equally across all days. What are the null and alternative hypotheses for this test?

Accepted Answer

Null: Accidents are equally likely on all days; Alternative: Accidents are not equally likely on all days.

Question 5

Given observed frequencies $$O_i$$ and expected frequencies $$E_i$$ for $$k$$ categories, what is the formula for the chi-square test statistic in a goodness-of-fit test?

Accepted Answer

$$\$$chi^2$$ = \sum_{i=1}^k \frac{(O_i - E_i)^2$}{E_i}$$

Question 6

In a two-way contingency table with $$r$$ rows and $$c$$ columns, what is the formula for the degrees of freedom for the chi-square test for independence?

Accepted Answer

$$df = (r-1)(c-1)$$

Question 7

A car manufacturer wants to know if brand loyalty is independent of the origin of the car (domestic, European, Asian). Which test should be used?

Accepted Answer

Chi-square test for independence

Question 8

In a two-way contingency table, what is the formula for the expected frequency $$E_{ij} in $$cell $$(i,j)$$?

Accepted Answer

$$E_{ij} = \frac{R_i C_j}{n}$$

Question 9

Suppose you conduct a chi-square test for independence and obtain a p-value of 0.03. If your significance level is 0.05, what is the correct conclusion?

Accepted Answer

Reject the null hypothesis; there is evidence of an association.

Question 10

A business analyst is comparing the mean sales of two independent stores. The sample sizes are small and population variances are unknown but assumed equal. Which test statistic should be used?

Accepted Answer

$$t=\frac{x_1$-x_2$-D_0$}{s_p \sqrt{\frac{1}{n_1$} + \frac{1}{n_2$}}}$$

Question 11

In a chi-square goodness-of-fit test with 4 categories, what is the degrees of freedom?

Accepted Answer

$$df = 3$$

Question 12

A survey of 948 cell phone users found 472 have one number, 331 have two, 111 have three, and 34 have four or more. The guide claims the proportions are 52%, 34%, 8%, and 6%. What is the expected frequency for the 'three numbers' category?

Accepted Answer

$$E = 948 	imes 0.08 = 75.84$$

Question 13

Which of the following is NOT an assumption of the chi-square test for independence in a contingency table?

Accepted Answer

The data are measured on an interval scale.

Question 14

A company wants to test if the distribution of customer complaints by type (billing, service, product, other) matches last year's distribution. What is the appropriate null hypothesis?

Accepted Answer

The distribution of complaints is the same as last year.

Question 15

If the calculated chi-square statistic is less than the critical value at the chosen significance level, what is the correct decision?

Accepted Answer

Fail to reject the null hypothesis.

Question 16

In a two-sample t-test for independent samples with equal variances, what is the formula for the pooled variance $$s_p^2$?

Accepted Answer

$$s_p^2$ = \frac{(n_1$-1)s_1$^2 + (n_2$-1)s_2$^2}{n_1$ + n_2$ - 2}$$

Question 17

A chi-square test for independence is performed on a 2x3 table. What is the degrees of freedom for this test?

Accepted Answer

$$df = (2-1)(3-1) = 2$$

Question 18

A researcher is testing whether the number of fatal DWI crashes is equally distributed across the week. The observed frequencies are: Sunday 41, Monday 24, Tuesday 25, Wednesday 28, Thursday 29, Friday 32, Saturday 38. What is the expected frequency for each day under the null hypothesis?

Accepted Answer

$$E = \frac{217}{7} = 31$$

Question 19

In a chi-square test for independence, which of the following would be evidence against the null hypothesis?

Accepted Answer

A large chi-square statistic and a small p-value

Business Statistics: Hypothesis Testing, Goodness-of-Fit, and Contingency Table Guidance

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