91Ó°ÊÓ

For Exercises 5 through \(20,\) perform these steps. a. State the hypotheses and identify the claim. b. Find the critical value(s). c. Compute the test value. d. Make the decision. e. Summarize the results. Use the traditional method of hypothesis testing unless otherwise specified. Game Attendance An athletic director suggests the median number for the paid attendance at 20 local football games is \(3000 .\) The data for a random sample are shown. At \(\alpha=0.05,\) is there enough evidence to reject the claim? If you were printing the programs for the games, would you use this figure as a guide? $$ \begin{array}{lllll}{6210} & {3150} & {2700} & {3012} & {4875} \\ {340} & {6127} & {2581} & {2642} & {2573} \\ {2292} & {2800} & {2500} & {3700} & {6030} \\ {5437} & {2758} & {3490} & {2851} & {2720}\end{array} $$

Step by step solution

01

State the Hypotheses

We start by stating the null and alternative hypotheses. The null hypothesis \(H_0\) is that the median attendance is 3000. This is the claim made by the athletic director. The alternative hypothesis \(H_a\) is that the median attendance is not 3000. Mathematically, these are expressed as: \( H_0: \text{median} = 3000 \) and \( H_a: \text{median} eq 3000 \).

02

Identify the Claim

The claim made by the athletic director is that the median attendance is 3000, so the claim is  associated with the null hypothesis, \(H_0\).

03

Find the Critical Values

Since we are testing the median, a sign test or Wilcoxon signed-rank test is appropriate. At \( \alpha = 0.05 \) with a two-tailed test, the critical values for the sign test can be found in relevant statistical tables. For a sample of 20, the critical value for a two-tailed sign test is based on the binomial distribution or obtained from a table, usually being around \( \pm z_{0.025} \), which is approximately \( \pm 1.96 \) for normal approximation.

04

Compute the Test Value

Compute the number of games with attendance greater than and less than 3000. We have 11 values greater than 3000 and 9 less than 3000. Using a sign test, the smaller value (9) becomes the test statistic. For a sign test with 20 observations, the test statistic follows a binomial distribution with parameters \(n = 20\) and \(p = 0.5\).

05

Make the Decision

Using the binomial distribution and the critical value, determine if the test statistic falls within the rejection region. Since 9 is not in the critical region for \( z \) approximation or direct table lookup, we fail to reject the null hypothesis. The decision is that there is not enough evidence to reject the claim that the median attendance is 3000.

06

Summarize the Results

To summarize, at a significance level of \( \alpha = 0.05 \), the test shows that there is not sufficient evidence to reject the athletic director's claim that the median attendance at the games is 3000. If tasked with printing programs, it would be reasonable to use 3000 as a guide for the median attendance.

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Key Concepts

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

Median

The median is a measure of central tendency that signifies the middle value in a data set. When organizing numbers in ascending order, the median represents a point at which half the numbers are greater and half are lesser. In this exercise, the athletic director claims that the median attendance is 3000.
The median is particularly useful when dealing with skewed data or outliers, as it is not as affected by these as the mean. For example, if a few of the games had an exceptionally high attendance, the mean would increase significantly, whereas the median would remain relatively unaffected. This attribute makes the median a preferable measure in various real-world scenarios, such as determining typical attendance at events without being distorted by extreme values.

Critical Value

Critical values are thresholds that determine the boundary for deciding whether to reject the null hypothesis in hypothesis testing. They depend on the chosen level of significance, often denoted by alpha (7), which represents the probability of making a Type I error.
For this exercise, we're dealing with a two-tailed test at a significance level of 0.05. The critical value is determined using the normal distribution table or binomial distribution, which, for a sample size of 20, closely approximates 9t 31.96 for a standard normal distribution.

• The critical value helps in forming the rejection region.
• If the test statistic falls outside this region, the null hypothesis is rejected.
• The choice of alpha influences the critical value and subsequently the hypothesis testing outcome.

Understanding critical values is integral to accurately interpreting results from hypothesis testing.

Sign Test

The sign test is a non-parametric method used to evaluate hypotheses about the median of a population. It assesses whether the number of data points above or below the median is significantly different.
For the sign test applied in this exercise:

• The null hypothesis assumes no difference, meaning values would equally fall above or below the median.
• We count the number of values above and below the claimed median of 3000.
• In our case, 11 values exceeded 3000, and 9 were below, resulting in a test statistic of 9.

This test is suitable when data does not conform to a normal distribution or when one wants to avoid strict assumptions typically associated with parametric tests. Since the test statistic lies within the non-rejection region, we maintain that insufficient evidence exists to refute the director's claim.

Wilcoxon Signed-Rank Test

The Wilcoxon signed-rank test is another non-parametric test used to assess whether the median of a single set of scores is equal to a known value. Unlike the sign test, it also considers the magnitude of differences, not just their signs.
In the context of this exercise, though not directly applied, the Wilcoxon signed-rank test could provide a more nuanced assessment, especially as it evaluates both direction and size of discrepancies from the hypothetical median. This could lead to more robust conclusions, particularly in samples where variabilities occur.

• With this test, each data point is ranked based on absolute differences from the median.
• Ranks are then signed and summed.
• The result is compared against a critical value from the Wilcoxon signed-rank table, indicating whether the median significantly deviates from the hypothesized value.

Though this test requires more calculations, it serves as a powerful alternative to further verify hypothesis conclusions when assumptions about the distribution of differences hold true.