Exercise Questions
Chapters 13, 14, and 17
James A Hardeman
Exercise 13-40
A suburban hotel derives its gross income from its hotel and restaurant operations. The
owners are interested in the relationship between the number of rooms occupied on a nightly basis and the revenue per day in the restaurant. Below is a sample of 25 days (Monday through Thursday) from last year showing the restaurant income and number of rooms occupied?
Use a statistical software package to answer the following questions.
a. Does the breakfast revenue seem to increase as the number of occupied room’s increases? Draw a scatter diagram to support your conclusion.
b. Determine the coefficient of correlation between the two variables. Interpret the value.
c. Is it reasonable to conclude that there is a positive relationship between revenue and occupied rooms? Use the .10 significance level.
d. What percent of the variation in revenue in the restaurant is accounted for by the number of rooms occupied?
Day Income Occupied Day Income Occupied
1 $1,452 23 14 $1,425 27
2 1,361 47 15 1,445 34
3 1,426 21 16 1,439 15
4 1,470 39 17 1,348 19
5 1,456 37 18 1,450 38
6 1,430 29 19 1,431 44
7 1,354 23 20 1,446 47
8 1,442 44 21 1,485 43
9 1,394 45 22 1,405 38
10 1,459 16 23 1,461 51
11 1,399 30 24 1,490 61
12 1,458 42 25 1,426 39
13 1,537 54
Exercise 14-26
26. A mortgage department of a large bank is studying its recent loans. Of particular interest
is how such factors as the value of the home (in thousands of dollars), education level oft he head of the household, age of the head of the household, current monthly mortgage payment (in dollars), and gender of the head of the household male=1, female = 0
relate to the family income. Are these variables effective predictors of the income of the household? A random sample of 25 recent loans is obtained.
Income Value Years of Mortgage
($ thousands) ($ thousands) Education Age Payment Gender
$40.3 $190 14 53 $230 1
39.6 121 15 49 370 1
40.8 161 14 44 397 1
40.3 161 14 39 181 1
40.0 179 14 53 378 0
38.1 99 14 46 304 0
40.4 114 15 42 285 1
40.7 202 14 49 551 0
40.8 184 13 37 370 0
37.1 90 14 43 135 0
39.9 181 14 48 332 1
40.4 143 15 54 217 1
38.0 132 14 44 490 0
39.0 127 14 37 220 0
39.5 153 14 50 270 1
40.6 145 14 50 279 1
40.3 174 15 52 329 1
40.1 177 15 47 274 0
41.7 188 15 49 433 1
40.1 153 15 53 333 1
40.6 150 16 58 148 0
40.4 173 13 42 390 1
40.9 163 14 46 142 1
40.1 150 15 50 343 0
38.5 139 14 45 373 0
a. Determine the regression equation.
b. What is the value of R 2? Comment on the value.
c. Conduct a global hypothesis test to determine whether any of the independent variables are different from zero.
d. Conduct individual hypothesis tests to determine whether any of the independent variables can be dropped.
e. If variables are dropped, recompute the regression equation and
Exercise 17 -22 Banner Mattress and Furniture Company wishes to study the number of credit applications received per day for the last 300 days. The information is reported on the next page.
Number of Credit Frequency
Applications (Number of Days)
0 50
1 77
2 81
3 48
4 31
5 or more 13
To interpret, there were 50 days on which no credit applications were received, 77 days
on which only one application was received, and so on. Would it be reasonable to conclude that the population distribution is Poisson with a mean of 2.0? Use the .05 significance level. Hint: To find the expected frequencies use the Poisson distribution with a mean of 2.0. Find the probability of exactly one success given a Poisson distribution with a mean of 2.0. Multiply this probability by 300 to find the expected frequency for the number of days in which there was exactly one application. Determine the expected frequency for the other days in a similar manner.
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To answer these exercise questions, you will need to use statistical software or a statistical calculator. Here is a breakdown of the steps to solve each exercise:
Exercise 13-40:
a. To determine if the breakfast revenue increases as the number of occupied rooms increases, you will need to create a scatter diagram. Use the number of rooms occupied as the independent variable (x-axis) and breakfast revenue as the dependent variable (y-axis). Plot the data points from the given sample and observe the trend.
b. To determine the coefficient of correlation between the two variables, you can use the statistical software to calculate the correlation coefficient (r). The correlation coefficient indicates the strength and direction of the linear relationship between the variables (+1 indicates a perfect positive linear relationship, -1 indicates a perfect negative linear relationship, and 0 indicates no linear relationship).
c. To determine whether there is a positive relationship between revenue and occupied rooms at the 0.10 significance level, you can conduct a hypothesis test. The null hypothesis (H0) is that there is no relationship (the correlation is zero) and the alternative hypothesis (Ha) is that there is a positive relationship (the correlation is greater than zero). Calculate the p-value and compare it to the significance level.
d. To determine the percentage of the variation in restaurant revenue accounted for by the number of rooms occupied, you can calculate the coefficient of determination (R^2). The coefficient of determination indicates the proportion of the total variation in the dependent variable (restaurant revenue) that can be explained by the independent variable (number of rooms occupied).
Exercise 14-26:
a. To determine the regression equation, you can use the statistical software to perform a multiple linear regression analysis. The regression equation will give you the formula to predict the income based on the values of the independent variables. It will include the regression coefficients for each variable.
b. The value of R^2 (coefficient of determination) indicates the proportion of the total variation in the income that can be explained by the independent variables. A higher R^2 value suggests a better fit of the regression model to the data.
c. To conduct a global hypothesis test to determine whether any of the independent variables are different from zero, you can use an analysis of variance (ANOVA) test. The null hypothesis (H0) is that none of the independent variables have a significant effect on the income, and the alternative hypothesis (Ha) is that at least one independent variable has a significant effect. Calculate the p-value and compare it to the significance level.
d. To conduct individual hypothesis tests to determine whether any of the independent variables can be dropped, you can calculate the t-statistic and p-value for each regression coefficient. The null hypothesis (H0) is that the regression coefficient is zero (no effect), and the alternative hypothesis (Ha) is that the regression coefficient is not zero. Compare the p-values to the significance level.
e. If variables are dropped, you will need to recompute the regression equation by excluding the dropped variables. The new equation will give you a revised formula to predict the income based on the remaining independent variables.
Exercise 17-22:
To determine if the population distribution is Poisson with a mean of 2.0, you can use a chi-square goodness-of-fit test. You will need to calculate the expected frequencies for each category under the assumption that the distribution is Poisson with a mean of 2.0. Then, calculate the chi-square test statistic and compare it to the critical value from the chi-square distribution with the appropriate degrees of freedom. If the test statistic is greater than the critical value, you can reject the null hypothesis and conclude that the population distribution is not Poisson with a mean of 2.0.
Remember to use the appropriate statistical software or calculator to perform the calculations for each exercise.