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A random sample of 99 students in a Conservatory of Music found that 9 of the 48 women sampled had "perfect pitch" (the ability to identify, without error, the pitch of a musical note), but only 1 of the 51 men sampled had perfect pitch. \({ }^{26}\) Conduct Fisher's exact test of the null hypothesis that having perfect pitch is independent of sex. Use a directional alternative and let \(\alpha=0.05 .\) Do you reject \(H_{0} ?\) Why or why not?

Short Answer

Expert verified
If the p-value obtained from Fisher's Exact Test is less than 0.05, we reject the null hypothesis, indicating that there is a statistical difference in having perfect pitch between women and men. The exact answer on whether to reject H0 depends on the p-value calculated.

Step by step solution

01

- State the Hypotheses

Define the null hypothesis (H0) and the alternative hypothesis (Ha). For this exercise, H0 is the hypothesis that having perfect pitch is independent of sex. The alternative hypothesis Ha is that there is a difference in having perfect pitch between the sexes, with the direction implied in the exercise suggesting women are more likely than men to have perfect pitch.
02

- Construct a Contingency Table

Create a 2x2 contingency table based on the data provided. Row labels will be 'Perfect Pitch' and 'No Perfect Pitch', and the column labels will be 'Women' and 'Men'. Fill the table with the data: women with perfect pitch (9), women without perfect pitch (48 - 9), men with perfect pitch (1), and men without perfect pitch (51 - 1).
03

- Calculate the Test Statistic

Use Fisher's Exact Test for assessing the null hypothesis. This test is especially suited for small sample sizes and is calculated based on the hypergeometric distribution. It's often done using software or a calculator because the calculations can become intensive.
04

- Determine the P-Value

Obtain the p-value associated with the observed contingency table. The p-value indicates the probability of obtaining a table as extreme as or more extreme than the one observed if the null hypothesis were true. Given that we're using a directional alternative hypothesis, ensure that the p-value is calculated for the one-sided test.
05

- Make a Decision

Compare the p-value to the significance level \( \alpha = 0.05 \). If the p-value is less than \( \alpha \), reject the null hypothesis. Otherwise, we do not have enough evidence to reject the null hypothesis.

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Key Concepts

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

Null Hypothesis
The null hypothesis is a statement used in statistical analysis that proposes there is no significant difference or effect. In the context of Fisher's exact test, the null hypothesis (\(H_0\)) claims that the two variables being studied are independent. For our exercise on perfect pitch and gender, the null hypothesis asserts that there is no relationship between one's sex and the ability to have perfect pitch. Therefore, the prevalence of perfect pitch in women and men should be the same. Establishing the null hypothesis is crucial as it forms the starting point of any statistical inference; it's what we test against to see if our data provides strong enough evidence to support an alternative hypothesis.
Contingency Table
A contingency table, or cross-tabulation, is a type of table in a matrix format that displays the frequency distribution of the variables. They are highly useful for examining the relationship between categorical variables. In our perfect pitch study, a 2x2 contingency table helps organize the data by two categories: sex (women and men) and the presence of perfect pitch. The four cells of the table correspond to the numbers of women and men with and without perfect pitch. The layout of this table is vital as it provides the foundation for most of the calculations required in Fisher's exact test, including the determination of the p-value.
P-Value
The p-value is a crucial concept in hypothesis testing. It represents the probability of observing a result as extreme as, or more extreme than, the results obtained during the test, assuming the null hypothesis is true. When we perform Fisher's exact test for our music conservatory exercise, the p-value tells us the likelihood of seeing the observed distribution of perfect pitch across sexes if, in fact, sex and perfect pitch are unrelated. A low p-value indicates that the observed data is unlikely under the null hypothesis and suggests that our observed difference (between the genders in this case) may be significant.
Significance Level
The significance level (\( \) often denoted by \( \alpha \) is the predetermined threshold against which we compare the p-value. It represents the risk of wrongly rejecting the null hypothesis (a false positive). In the case of our exercise, the significance level is set at 0.05 or 5%. This means if our p-value is less than 0.05, there's less than a 5% chance that the observed difference in perfect pitch between sexes is due to random chance, and we would reject the null hypothesis. Deciding on an appropriate significance level before conducting a test is key to the integrity of hypothesis testing.
Perfect Pitch
Perfect pitch, also known as absolute pitch, is the rare ability to identify or recreate a musical note without any reference tone. It's a fascinating phenomenon that's quite useful for musicians, and it's often thought to be either inherited, or developed at a very young age. In the context of our exercise, we're interested in testing whether this ability is equally distributed across sexes or whether one gender is statistically more likely to possess perfect pitch.
Statistical Independence
Statistical independence refers to a situation where the occurrence of one event does not affect the probability of occurrence of another event. For example, flipping a coin is independent of rolling a die. In our case, if perfect pitch and sex are independent, knowing someone's sex gives no information about their likelihood of having perfect pitch. Fisher's exact test examines the independence of two categorical variables, like sex and perfect pitch, by viewing the probabilities given in a well-constructed contingency table. When a significant relationship is found, we can surmise that the variables are not independent of one another.

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