Question

From the website Golf.com, part of Sports Illustrated Sites, we obtained the scores for the first and second rounds of the \(2013\) U.S Open golf tournament. You will find those scores on the Weiss Stats site.

• For the estimation and predictions, use a first-round score of \(72\)
• For the correlation test, decide whether first-round and second round scores are positively linearly correlated.

a. determine the sample regression equation.

b. find and interpret the standard error of the estimate.

c. decide at the \(5%\) significance level, whether the data provide sufficient evidence to conclude that the predictor variable is useful for predicting the response variable.

d. determine and interpret a point estimate for the conditional mean of the response variable corresponding to the specified value of the predictor variable.

e. find and interpret a \(95%\) confidence interval for the conditional mean of the response variable corresponding to the specified value of the predictor variable.

Short answer

Part a. \(\hat{y}=56.9171+0.2275x\)

Part b. \(s_{e}=3.0599\)

Part c. The null hypothesis is rejected, so the consumption of beef last year is less compared to year \(2011\).

Part d. \(\hat{y}=73.2971\)

Part e. The test is invalid, on the grounds that there are exceptions are available in the informational index and these anomalies impact the finish of the test (while it isn't proper to expel the information esteem from the informational index).

Step-by-step solution

Part a. Step 1. Given information

The value of a and b is given

\(a\approx 56.9171\)

\(b\approx 0.2275\)

Part a. Step 2. Calculation

We know that the regression line

\(\hat{y}=a+bx\)

Put the given value of a and b into the above equation

\(\hat{y}=a+bx\)

\(hat{y}=56.9171+0.2275x\)

The solution is \(hat{y}=56.9171+0.2275x\)

Part b. Step 1. Calculation

We know that the regression line

\(\hat{y}=a+bx\)

Put the given value of a and b into the above equation

\(\hat{y}=a+bx\)

\(hat{y}=56.9171+0.2275x\)

Calculate the standard error

\(s_{e}=\frac{\sigma}{\sqrt{n}}\)

After solving we will get

\(s_{e}=3.0599\)

The solution is \(s_{e}=5.1717\)

Part c. Step 1. Calculation

The predictor variable useful for predicting the response variable, when the slope of the sample regression line is nonzero.

\(H_{0}:\beta =0\)

\(H_{a}:\beta \neq 0\)

The p-value of the hypothesis will be

\(P\approx 0.0072\)

The null hypothesis is rejected.

\(P<0.05 \rightarrow\) Fail to Reject \(H_{0}\)

The null hypothesis is not rejected, so there is no sufficient evidence are provided for the sample.

Part d. Step 1. Calculation

We know that the linear regression relation from part (a)

\(\hat{y}=56.9171+0.2275x\)

Put \(x=72\)

Then calculate the conditional mean

\(\hat{y}=56.9171+0.2275\times 72\)

\(\hat{y}=73.2971\)

Program:

Query:

• First, we have defined \(75\) random samples.

• Sketch the graph of normal probability plot, boxplot, histogram and stem-and-leaf plot.

Part e. Step 1. Calculation

Calculate the mean of the data of beef consumption this year

\(\bar{x}=\frac{\sum _{i-1}^{n}x_{i}}{n}\)

\(=\frac{92+80+...+35+12}{40}\)

\(=\frac{2670}{40}\)

\(\bar{x}=66.75\)