91Ó°ÊÓ

The overall distance traveled by a golf ball is tested by hitting the ball with Iron Byron, a mechanical golfer with a swing that is said to emulate the distance hit by the legendary champion Byron Nelson. Ten randomly selected balls of two different brands are tested and the overall distance measured. $$\begin{aligned}&\text { The data follow: }\\\&\text { Brand } 1: 275,286,287,271,283,271,279,275,263,267\\\&\text { Brand 2: } 258,244,260,265,273,281,271,270,263,268\end{aligned}$$ a. Is there evidence that overall distance is approximately normally distributed? Is an assumption of equal variances justified? b. Test the hypothesis that both brands of ball have equal mean overall distance. Use \(\alpha=0.05 .\) What is the \(P\) -value? c. Construct a \(95 \%\) two-sided \(\mathrm{CI}\) on the mean difference in overall distance for the two brands of golf balls. d. What is the power of the statistical test in part (b) to detect a true difference in mean overall distance of 5 yards? e. What sample size would be required to detect a true difference in mean overall distance of 3 yards with power of approximately \(0.75 ?\)

Step by step solution

01

Check Normality

To evaluate whether the distances are normally distributed, we can use graphical methods (such as Q-Q plots) or statistical tests like the Shapiro-Wilk test. Given the sample sizes, visual inspection through Q-Q plots would suffice for a preliminary assessment.

02

Check Equal Variances

Use Levene's Test or Bartlett's Test to evaluate if the variances of the two brands are equal. These tests examine the null hypothesis that the variances are homogeneous.

03

Formulate Hypotheses

State the null hypothesis as \( H_0: \mu_1 = \mu_2 \) and the alternative hypothesis as \( H_a: \mu_1 eq \mu_2 \), where \( \mu_1 \) and \( \mu_2 \) are the true mean distances for Brand 1 and Brand 2.

04

Perform t-Test for Equality of Means

Since both sample sizes are 10, use a two-sample t-test. If variances are assumed equal, use a pooled variance t-test. Otherwise, use Welch's t-test. Calculate the test statistic and compare it to the critical t-value at \( \alpha = 0.05 \). Also determine p-value from the test.

05

Construct Confidence Interval

Using the results from the t-test, calculate a 95% confidence interval for the difference in means. This involves the estimated difference in means, the standard error, and the critical value from a t-distribution.

06

Calculate Power of the Test

To calculate the power, determine the non-centrality parameter under the alternative hypothesis that there's a 5-yard difference in means. Use the statistical software or power analysis handbooks that provide power calculation formulas for t-tests.

07

Determine Required Sample Size

Using desired power (0.75) and effect size (3 yards difference), perform a sample size calculation. Use the formula for sample size calculation in two-sample t-tests or consult software that performs this calculation explicitly.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Normal Distribution

Normal distribution is a foundational concept in statistics. It describes how data points, such as those measuring the distance of golf balls, are spread out.
It's a bell-shaped curve that extends infinitely in both directions. The data is symmetrically distributed around the mean.
To check if the golf ball distances are normally distributed, we use methods like Q-Q plots or the Shapiro-Wilk test. These methods help visualize whether the data fits the normal distribution.
Normality is important because many statistical tests, like the t-test, rely on this assumption to be valid. When you see a Q-Q plot, you expect the points to form a straight line. If they do, this suggests that the data is normally distributed. If the points deviate greatly, the data might not be normal.
Visual methods like these are useful, especially with smaller sample sizes, because they offer a quick glance at the distribution.

t-test

The t-test is a statistical test used to compare the means of two groups, like the two brands of golf balls.
It helps determine if there's a significant difference between the groups' means.We first set up hypotheses:

• Null hypothesis (\( H_0 \)): The means are equal (\( \mu_1 = \mu_2 \)).
• Alternative hypothesis (\( H_a \)): The means are not equal (\( \mu_1 e \mu_2 \)).

The type of t-test you use depends on the assumption of equal variances:

• Pooled variance t-test if variances are assumed equal.
• Welch's t-test if variances are not assumed equal.

To perform the test, calculate the test statistic and compare it with the critical t-value at \( \alpha = 0.05 \) (5% significance level).
Also, find the p-value, which tells you the probability of observing your results, or more extreme results, if the null hypothesis is true.

Confidence Interval

A confidence interval (CI) offers a range of values that likely includes the true difference between population means.
For the golf ball data, you'd construct a 95% CI to understand how much the mean distances differ.To calculate it, you need:

• The estimated difference in means.
• The standard error of the difference.
• The critical value from the t-distribution, based on your confidence level.

Once you have these, the confidence interval is calculated as:\[(\text{Difference} \, \pm \, \text{Critical Value} \, \times \, \text{Standard Error})\]This tells you the range, and if the CI does not include 0, it suggests a significant difference in means.
Confidence intervals are valuable because they give not just a point estimate, but a range, reflecting uncertainty in the estimate.

Sample Size Calculation

Sample size calculation is crucial to ensure your study has enough power to detect a meaningful difference.
In this golf ball scenario, it's about determining how many golf balls to test to detect a true difference in means. The sample size you need depends on factors like:

• Your desired power (e.g., 0.75 in this case).
• The expected effect size (e.g., a 3-yard difference).
• The level of significance (e.g., 0.05).

To calculate sample size, use formulas specific to two-sample t-tests or software that can handle these computations.
Achieving the correct sample size is vital because a small sample might fail to detect a true effect, whereas a very large sample might waste resources without providing additional value.