Question

Simple Linear Regression Problem

QUESTION 6 The state office of Australia Post believed that mail throughput at suburban post offices increases with the number of employees. From a random sample of twelve major suburban post offices, the following regression equation was estimated: Y =-2.59 + 78.15 X Standard error of the estimate: 238.54 Sum of squares of X: 112.5 where Y is the average number of mail packages processed per hour and X is the number of effective full-time employees working at the post office 1. State the direction of the alternative hypothesis used to test the state office's belief. Type gt (greater than), ge (greater than or equal to), It (less than), le (less than or equal to) or ne (not equal to) as appropriate in the box 2. Use the tables in the textbook to determine the critical value used to conduct the test, assuming a 1 % level of significance. If doing a two tail test, state only the positive value to three decimal places 3. Calculate the test statistic, reporting your answer to two decimal places. 4. Is the null hypothesis rejected for this test? Type yes or no. 5. Disregarding your answer in part 4, if the null hypothesis was rejected, does a larger number of employees in a post office suggest it will have a greater throughput of mail? Type yes or no.

Answer 1

the regression equation to be estimated is

$Y=\beta_0+\beta_1X+\epsilon$

where Y is the average number of mail packages processed per hour

X is the number of effective full time employees working at the post office.

$\beta_0$ is the intercept of the regression line

$\beta_1$ is the slope coefficient of the line

$\epsilon \stackrel{iid}\sim \mathcal{N}(0,\sigma^2)$ is a random error

The estimated line is

$\hat{Y}=-2.59+78.15X$

the estimated value of the intercept is

$\hat{\beta}_0=-2.59$

The estimated value of the slope is

$\hat{\beta}_1=78.15$

The standard error of the estimates is

$s=238.54$

The sum of square of X is

$SS_x=112.5$

The standard error of slope estimate is

$s(\hat{\beta}_1)=\frac{s}{\sqrt{SS_x}}=\frac{238.54}{\sqrt{112.5}}=22.49$

1. The state office of Australia believed that mail throughput at suburban office increases with the number of employees.

That means in the regression equation

$Y=\beta_0+\beta_1X+\epsilon$

the mail throughput Y would increase with X (the number of employees) if the slope coefficient $\beta_1>0$

This would be our alternative hypothesis.

the hypotheses we want to test are

$\begin{align*} &H_0:\beta_1=0\leftarrow\text{null hypothesis: Number of employees has no effect on throughput}\\ &H_a:\beta_1>0\leftarrow\text{alternative hypothesis: Throughput increases with the number of employees}\\ \end{align*}$

ans: The direction of the alternative hypothesis used to test the state office's belief is gt

2) this is a right tail test (The alternative hypothesis has ">").

The number of observations in the sample is n=12

The degrees of freedom for t statistics is n-2=12-2=10

For significance level $\begin{align*} \alpha=0.01 \end{align*}$ , the right tail critical value is $\begin{align*} P(T>t_{\alpha})=0.01 \end{align*}$

Using the t distribution table in the text book we get for right tail area of 0.01 and df=10, the critical value of t is 2.764

ans: The critical value is 2.764

3. The hypothesized value of the slope is $\begin{align*} \beta_{H_0}=0 \end{align*}$ (From the null hypothesis)

The test statistics is

$\begin{align*} t=\frac{\hat{\beta}_1-\beta_{1H_0}}{s(\hat{\beta}_1)}=\frac{78.15-0}{22.49}=3.47 \end{align*}$

ans: the test statistics is 3.47

4) We will reject the null hypothesis if the test statistics is greater than the critical value.

Here the test statistics is 3.47 and it is greater than the critical value of 2.764. Hence we reject the null hypothesis.

ans: Is the null hypothesis rejected for this test? Yes

5) Since we reject the null hypothesis, we can conclude that there is sufficient evidence to support the state office's belief that mail throughput at suburban office increases with the number of employees.

ans: yes