Question

Grades People Office 365 My Media Media Gallery A grocery store manager did a study to look at the relationship between the amount of time (in minutes) customers spend in the store and the amount of money (in dollars) they spend. The results of the survey are shown below. Accou nt Time 24 10 14 27151923 15 Money 100 57 35 97 68 57 83 50 Dashb Library oard 0 Policies Make Tu- Coursition es Payment Accredible Calen dar Inbox Help a. Find the correlation coefficient: Round to 2 decimal places. b. The null and alternative hypotheses for correlation are: Ho:?V=0 HI +0 The p-value is: (Round to four decimal places) c. Use a level of significance of a = 0.05 to state the conclusion of the hypothesis test in the context of the study. - There is statistically insignificant evidence to conclude that a customer who spends more time at the store will spend more money than a customer who spends less time at the store. There is statistically significant evidence to conclude that a customer who spends more time at the store will spend more money than a customer who spends less time at the store. . There is statistically insignificant evidence to conclude that there is a correlation between the amount of time customers spend at the store and the amount of money that they spend at the store. Thus, the use of the regression line is not appropriate. .There is statistically significant evidence to conclude that there is a correlation between the amount of time customers spend at the store and the amount of money that they spend at the store. Thus, the regression line is useful d. 2 - (Round to two decimal places) e. Interpret 2: .69% of all customers will spend the average amount of money at the store. .There is a 69% chance that the regression line will be a good predictor for the amount of money spent at the store based on the time spent at the store. Given any group that spends a fixed amount of time at the store, 69% of all of those customers will spend the predicted amount of money at the store. .There is a large variation in the amount of money that customers spend at the store, but if you only look at customers who spend a fixed amount of time at the store, this variation on average is reduced by 69%. 1. The equation of the linear regression lines: (Please show your answers to two decimal places) 8. Use the model to predict the amount of money spent by a customer who spends 14 minutes at the store. Dollars spent (Please round your answer to the nearest whole number.)

Answer 1

X: Time and Y: Money

a) Correlation coefficient

The formula for correlation coefficient(r)

$r = \frac{n\sum xy-\sum x\sum y}{\sqrt{[n\sum x^2-(\sum x)^2][n\sum y^2-(\sum y)^2]}}$

where n = number of pairs = 8

$\sum x = 147, \sum y = 547, \sum x^2 = 2941, \sum y^2 = 41145, \sum xy = 10841$

$r=\frac{8*10841-(147*547)}{\sqrt{[8*2941-147^2][8*41145-547^2]}} = 0.83$

Correlation coefficient, r = 0.83

b) The null and alternative hypothesis for correlation

$H_0: \rho =0 \ and \ H_1: \rho \neq 0$

The t test sttaistics for the test is,

$t = r\sqrt{\frac{n-2}{1-r^2}} = 0.83\sqrt{\frac{8-2}{1-0.83^2}}=3.645$

Degrees of freedom = n - 2 = 6 and the test is two-tailed test.

P-value using the excel function is

=tdist(test statistics, df, tail of test) = tdist(3.645,6,2) = 0.0108

P-value = 0.0108

c) alpha = 0.05 and p-value = 0.0108

If P-value > alpha thne fail to reject the null otherwise reject the null hypothesis.

Here p-value is not more than alpha so reject the null hypothesis that is there is correlation between both variables thus the regression line is useful for prediction.

Last option is correct.

d) r²

Square of r is nothing but r^2

that is 0.69

r²   = 69%

e) Interpretation of r²

Option second is correct for interpretation.

f) Equation of linear regression line

$\hat y = a+bx$

where a is intercept and b is slope

The formula of slope is

$b = \frac{n\sum xy-\sum x\sum y}{n\sum x^2-(\sum x)^2} = \frac{8*10841-147*547}{8*2941-147^2} = 3.29$

The formula of intercept is

$a = \frac{\sum y-b\sum x}{n} = \frac{547-3.29*147}{8} = 7.92$

The equation of line is

$\hat y = 7.92+3.29x$

g) Predict amount of money spent.

x = 14 minutes

$\hat y = 7.92+3.29x=7.92+3.29*14 = 53.98$

Approximately 54 dollars

Dollar spent = 54