A local newspaper randomly selects 20 patrons of the...

A local newspaper randomly selects 20 patrons of the Springwood Restaurant on a given Saturday night and has each patron rate the quality of his or her meal as 5 (excellent), 4 (good), 3 (average), 2 (poor), or 1 (unsatisfactory). When the results are summarized, it is found that there are 16 ratings of 5,3 ratings of 4 and 1 rating of $3 .$ Let $M_{d}$ denote the population median rating that would be given by all possible patrons of the restaurant on the Saturday night. a. Test $H_{0}: M_{d}=4.5$ versus $H_{a}: M_{d}>4.5$ by setting $\alpha=.05$ b. Reason that your conclusion in part $a$ implies that we have very strong evidence that the median rating that would be given by all possible patrons is $5 .$

A local newspaper randomly selects 20 patrons of the Springwood Restaurant on a given Saturday night and has each patron rate the quality of his or her meal as 5 (excellent), 4 (good), 3 (average), 2 (poor), or 1 (unsatisfactory). When the results are summarized, it is found that there are 16 ratings of 5,3 ratings of 4 and 1 rating of $3 .$ Let $M_{d}$ denote the population median rating that would be given by all possible patrons of the restaurant on the Saturday night.
a. Test $H_{0}: M_{d}=4.5$ versus $H_{a}: M_{d}>4.5$ by setting $\alpha=.05$
b. Reason that your conclusion in part $a$ implies that we have very strong evidence that the median rating that would be given by all possible patrons is $5 .$

Key Concepts

Significance Level (alpha)

The significance level, often denoted as alpha (?), is the probability threshold used in hypothesis testing to decide whether to reject the null hypothesis. A commonly used alpha of 0.05 indicates a 5% risk of committing a Type I error, which is rejecting the null hypothesis when it is actually true.

Sign Test

The sign test is a simple nonparametric method used to test claims about a population median. It involves counting the number of observations that exceed or fall below a hypothesized median without making assumptions about the shape of the distribution, and it uses the binomial distribution to assess the significance of the observed counts.

Evidence in Hypothesis Testing

In hypothesis testing, evidence is evaluated by comparing the test statistic or p-value computed from the data against the predetermined significance level. If the test statistic falls into the rejection region defined by alpha, or if the p-value is less than alpha, then there is strong evidence against the null hypothesis in favor of the alternative hypothesis.

Median

The median is a measure of central tendency that divides a dataset into two equal parts, with half the observations above and half below. When evaluating a population’s central tendency, the median is especially useful in skewed distributions or with ordinal data, as it is less affected by outliers compared to the mean.

Hypothesis Testing

Hypothesis testing is a statistical process used to decide whether there is enough evidence in a sample of data to infer that a certain condition is true for the entire population. It involves setting up a null hypothesis, which typically represents no effect or a baseline claim, and an alternative hypothesis that represents the claim being tested against the null.

Nonparametric Statistics

Nonparametric statistics provide methods for analyzing data that do not assume a specific parametric form for the population distribution. This is particularly useful when dealing with ordinal data or when the underlying distribution is unknown, making methods such as the sign test appropriate for hypothesis testing concerning medians.

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