91Ó°ÊÓ

Each of a group of 20 intermediate tennis players is given two rackets, one having nylon strings and the other synthetic gut strings. After several weeks of playing with the two rackets, each player will be asked to state a preference for one of the two types of strings. Let \(p\) denote the proportion of all such players who would prefer gut to nylon, and let \(X\) be the number of players in the sample who prefer gut. Because gut strings are more expensive, consider the null hypothesis that at most \(50 \%\) of all such players prefer gut. We simplify this to \(H_{0}: p=.5\), planning to reject \(H_{0}\) only if sample evidence strongly favors gut strings. a. Which of the rejection regions \(\\{15,16,17,18\), \(19,20\\},\\{0,1,2,3,4,5\\}\), or \(\\{0,1,2,3,17,18\), \(19,20\\}\) is most appropriate, and why are the other two not appropriate? b. What is the probability of a type I error for the chosen region of part (a)? Does the region specify a level \(.05\) test? Is it the best level \(.05\) test? c. If \(60 \%\) of all enthusiasts prefer gut, calculate the probability of a type II error using the appropriate region from part (a). Repeat if \(80 \%\) of all enthusiasts prefer gut. d. If 13 out of the 20 players prefer gut, should \(H_{0}\) be rejected using a significance level of . 10?

Step by step solution

01

Identify Null and Alternative Hypotheses

The null hypothesis is that half or fewer than half of the players prefer gut strings. We express this as \(H_0: p = 0.5\). The alternative hypothesis, suggesting that more than half prefer gut, is \(H_A: p > 0.5\).

02

Determine the Appropriate Rejection Region

We need to choose the rejection region that is appropriate. The standard approach is to consider rejecting \(H_0\) when there is strong evidence against it. Since we want to reject \(H_0\) only if the evidence strongly favors gut, \{15, 16, 17, 18, 19, 20\} would be the appropriate region as it indicates that a majority significantly larger than 50% prefer gut. The region \{0,1,2,3,4,5\} suggests extreme preference for nylon, and \{0,1,2,3,17,18,19,20\} mixes preferences, making it unsuitable for a one-sided test favoring gut.

03

Type I Error Probability

The Type I error occurs when we reject \(H_0\) even though it's true. If we select the region \{15,16,17,18,19,20\}, we must calculate the probability of \(X\) taking on one of these values, given \(p = 0.5\). Using the binomial distribution, \[ P(X \in \{15,16,17,18,19,20\}) = \sum_{k=15}^{20} \binom{20}{k} (0.5)^k (0.5)^{20-k} \]. This sum should be less than or equal to 0.05 for it to be a level 0.05 test, which it approximately satisfies.

04

Probability of a Type II Error

A Type II error is not rejecting \(H_0\) when \(H_A\) is true. For \(p = 0.6\), calculate \[ P(X < 15) = \sum_{k=0}^{14} \binom{20}{k} (0.6)^k (0.4)^{20-k} \]. Repeat for \(p = 0.8\) using \(0.8^k\) and \(0.2^{20-k}\) to determine the probability \(P(X < 15)\).

05

Hypothesis Testing With Given Data

With 13 out of 20 preferring gut, we calculate the z-score for \(X = 13\): \[ z = \frac{13 - 10}{\sqrt{20\cdot 0.5 \cdot 0.5}} = \frac{3}{\sqrt{5}} \].The critical value for a significance level of 0.10 is approximately 1.28 (one-tailed). Compute \(z \approx 1.34\), which is greater than 1.28, suggesting a rejection of \(H_0\) at \(\alpha = 0.10\).

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Key Concepts

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

Type I Error

In hypothesis testing, a Type I error occurs when we reject the null hypothesis ( H_0 ) even though it is true. Essentially, it's a false alarm. Here, we're saying there's an effect or a preference when, in fact, there isn't. This is similar to telling someone that an alarm has sounded when there's actually no fire.

In the context of our tennis racket exercise:

• The null hypothesis ( H_0: p = 0.5 ) proposes that 50% or fewer players prefer gut strings.
• A Type I error would mean concluding that more than 50% of players prefer gut strings when they actually don't.
• When using a rejection region like {15,16,17,18,19,20}, we calculate the probability of committing a Type I error at the level of significance (often set at 0.05 in tests).

Understanding Type I error is crucial, as it helps researchers set accurate confidence levels in their conclusions. The core task is to minimize these errors, ideally balancing them against Type II errors without overly compromising either.

Type II Error

A Type II error in hypothesis testing occurs when we fail to reject the null hypothesis ( H_0 ) when, in fact, the alternative hypothesis ( H_A ) is true. This type of error is essentially a missed detection. You mistakenly say that the party isn't happening, even though it's in full swing.

For the tennis racket problem:

• If we hypothesize that more than half of the players (say 60% or 80%) prefer gut strings and fail to reject the null hypothesis, a Type II error has occurred.
• Calculations for a Type II error involve determining the probability of not rejecting H_0 when these proportions are true. For instance, when 60% prefer gut, the probability P(X < 15) is computed.

Type II errors are critical to understand because they relate directly to the power of a test. The power is denoted as 1 minus the probability of a Type II error, reflecting how well a test can detect an actual effect.

Null and Alternative Hypotheses

Hypothesis testing always begins with formulating the null and alternative hypotheses. These hypotheses form the foundation of statistical testing, providing the baseline and alternative claims to be tested.

In our tennis string exercise:

• The null hypothesis ( H_0: p = 0.5 ) suggests that half or fewer of the players prefer gut strings over nylon strings.
• The alternative hypothesis ( H_A: p > 0.5 ) proposes that more than half of the players prefer gut strings.

Setting these hypotheses frames the direction and nature of the test, directing researchers on whether to look for evidence to support or refute H_0 . They act as a guiding checkpoint on whether a theory should be supported or if the opposite claim gains merit.

Rejection Region

The rejection region in hypothesis testing is the range of values for which you will reject the null hypothesis ( H_0 ). It's a crucial concept, as it defines the cutoff point where evidence becomes strong enough to say "something notable is happening."

For our exercise:

• Given that we are testing if more than 50% prefer gut strings, a reasonable rejection region would include outcomes with a strong lean towards gut preference, such as {15, 16, 17, 18, 19, 20}.
• Choosing this endpoint helps establish clear criteria: if the count of players who prefer gut strings falls in this cluster, the null hypothesis is rejected.
• Correctly setting the rejection region ensures the decisions made detach from randomness, focusing instead on evidence-based deviance.

Using rejection regions effectively aids in reducing uncertainty in hypothesis testing, allowing one to draw concrete, statistically backed conclusions.