Question

n a survey from 1998, 449 teenagers were surveyed about the music that they listen to. Of these teenagers, 129 of them said that their favorite genre of music is hip-hop. In a similar survey from 2008, 176 of 509 teenagers surveyed said that their favorite genre is hip-hop. Use a two-proportion hypothesis test to determine whether the proportion of teenagers whose favorite genre of music is hip-hop has changed from 1998 to 2008. Assume that the samples are random and independent. Use α=0.01. Let the sample from 1998 correspond to sample 1 and the other to sample 2. (a) Which answer choice shows the correct null and alternative hypotheses for this test?

Answer 1

The correct null and alternative hypotheses for this test are:

Null hypothesis (H₀): The proportion of teenagers whose favorite genre of music is hip-hop is the same in 1998 and 2008.

Alternative hypothesis (H₁): The proportion of teenagers whose favorite genre of music is hip-hop is different in 1998 and 2008.

Step-by-step explanation:

We are interested in testing if there is a difference in the proportions of teenagers who prefer hip-hop in the two samples. Therefore:

H₀ assumes the proportions are equal: We assume that the observed difference in the samples could be due to chance variations, and there is no actual difference in the population proportions.

H₁ allows for a difference: This hypothesis states that there is a significant difference in the true population proportions of teenagers who prefer hip-hop between 1998 and 2008.

The other answer choices either state the hypotheses in the wrong direction (e.g., H₀ stating a difference), or focus on the specific proportions observed in the samples (the test is about population proportions, not sample proportions).

Therefore, the chosen answer choice correctly represents the null and alternative hypotheses for testing the change in the proportion of teenagers who prefer hip-hop across the two survey years.

Answer 2

We accept H₀

Step-by-step explanation:

To compare two proportion:

proportion 1. 1998

n₁ = 449

p₁ = 129/449 p₁ = 0,2873

proportion 2. 2008

n₂ = 509

p₂ = 176/509 p₂ = 0,3457

Test hypothesis

Null Hypothesis H₀ p₂ - p₁ = 0

Alternative Hypothesis Hₐ p₂ > p₁

We have a one tail test ( to the right)

As α = 0,01 we find in z-table the z score z = 2,29 ( approximation for 0,011)

To calculate z(s),

z(s) = ( p₂ - p₁ ) /√ p*q ( 1/n₁ + 1/n₂)

p = ( x₁ + x₂ ) / n₁ + n₂

p = 129 + 176 / 449 +509

p = 0,3184 and q = 1 - p q = 0,6816

1/n₁ + 1/n₂ = 1/ 449 + 1 / 509 = 0,00222 + 0,001964

1/n₁ + 1/n₂ = 0,004184

z(s) = 0,0584/ √(0,3184*0,6816)*0,004184

z(s) = 0,0584/ 0,03013

z(s) = 1,938

z(s) < z(c) 1,938 < 2,29

z(s) is in the acceptance region we accept H₀ at α = 0,01