T and S share a digital music player that randomly selects songs to play.
Together T and S have loaded 3476 songs.
T and S want to know if each have loaded a different proportion of songs.
When in random-selection mode, 22 of first 30 songs selected were loaded by S.
Let p denote proportion of songs loaded by S.
State the null and alternative hypotheses to be tested.
B) How strong is evidence that T and S have each loaded a different proportion of songs?
B) To test whether T and S have each loaded a different proportion of songs, we can set up the null and alternative hypotheses as follows:
Null Hypothesis (H0): The proportion of songs loaded by T is equal to the proportion of songs loaded by S. (pT = pS)
Alternative Hypothesis (Ha): The proportion of songs loaded by T is not equal to the proportion of songs loaded by S. (pT ≠ pS)
We will use a hypothesis test to determine if the evidence supports rejecting the null hypothesis in favor of the alternative hypothesis.
B) In order to test the hypothesis that T and S have each loaded a different proportion of songs, we can conduct a two-proportion z-test.
The null hypothesis (H₀) will state that the proportions of songs loaded by T and S are equal, while the alternative hypothesis (H₁) will state that the proportions are different.
Null Hypothesis (H₀): The proportion of songs loaded by T is equal to the proportion of songs loaded by S.
Alternative Hypothesis (H₁): The proportion of songs loaded by T is different from the proportion of songs loaded by S.
To assess the strength of evidence that T and S have loaded different proportions of songs, we can calculate the test statistic and its associated p-value. The test statistic for a two-proportion z-test can be calculated using the following formula:
z = (p̂₁ - p̂₂) / sqrt((p̂₁ * (1 - p̂₁) / n₁) + (p̂₂ * (1 - p̂₂) / n₂))
where p̂₁ and p̂₂ are the sample proportions of songs loaded by T and S, n₁ and n₂ are the corresponding sample sizes, respectively.
Once the test statistic is calculated, we can find the p-value associated with it. The p-value represents the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true. A lower p-value indicates stronger evidence against the null hypothesis, favoring the alternative hypothesis.
By comparing the calculated test statistic to a critical value (typically based on a significance level), we can make a decision on whether to reject or fail to reject the null hypothesis, providing a conclusion regarding the strength of evidence for each scenario.