Question

Long Drives. The Professional Golfer's Association of America (PGA) organizes golf tournaments for professional golfers. The following table lists the longest drives, in yards, recorded during a PGA tournament for a random sample of $26$ golfers. Use the technology of your choice to decide whether applying the $z-$interval procedure to these data is reasonable. Explain your answer.

Short answer

The applying $z-$interval procedure to the given data is not reasonable.

Step-by-step solution

Given information

395	400	377	367	386	407	383
396	371	376	373	384	369	391
386	393	374	366	388	371	416
375	450	370	379	381

Explanation

Check whether using the z-interval technique on the given data is appropriate.

Draw the normal probability plot for golfers using MINITAB.

Procedure for MINITAB:

Step 1: Select Probability Plot from the Graph menu.

Step 2: Click OK after selecting Single.

Step 3: In the Graph variables section, type Golfers in the column.

Step 4: Click the OK button.

OUTPUT FROM MINITAB:

Draw the histogram for golfers using MINITAB.

Procedure for MINITAB:

Step 1: Select Histogram from the Graph menu.

Step 2: Click OK after selecting Simple.

Step 3: In the Graph variables section, type in the Golfers column.

Step 4: Click the OK button.

OUTPUT FROM MINITAB:

Explanation

The following are the conditions for using the $z-$interval procedure:

Small Sample size:

The $z-$interval approach is employed when the sample size is less than 15 and the variable is normally distributed or extremely near to being normally distributed.

Moderate Sample size:

When the sample size is between $15$and $30$and the variable is not normally distributed or there is no outlier in the data, the $z-$interval technique is performed.

Large Sample size:

The $z-$interval technique is utilized without restriction if the sample size is bigger than $30$

The distribution of the variable is extremely skewed, and the sample size $(=26)$ is lower, as shown by the MINITAB result. As a result, the distribution of Golfers deviates from the normal. As a result, applying the $z-$interval technique to the given data is illogical.