Do these data provide evidence that the response to the television show is different for boys and girls? The producer of a new children’s television show arranges for a screening of the show for a random sample of 500 children. At the end of the viewing, the children were asked whether they were not interested in the show, somewhat interested in the show or very interested in the show. The results were classified according to gender and shown in the table below.

The data doesn't provide sufficient evidence that the response to the television show is different for boys and girls

Step-by-step explanation:

We will be conducting a χ²-test of independence where the conditions are:

Expected value is >5 (condition is met)

Random sample (condition is met)

Categorical data (condition is met

Our null and alternate hypotheses are:

H: Observed=Expected

Ha: Observed≠Expected

Our expected values would be:

Not Interested + Boys : (220)(110)/500 = 48.4

Not Interested + Girls : (280)(110)/500 = 61.6

Somewhat Interested + Boys : (220)(205)/500 = 90.2

Somewhat Interested + Girls : (280)(205)/500 = 114.8

Very Interested + Boys: (220)(185)/500 = 81.4

Very Interested + Girls: (280)(185)/500 = 103.6

Now we will use the formula χ² = ∑[(O-E)^2/E] where O is the observed value and E is the expected value:

χ² = (50-48.4)²/48.4 + (60-61.6)²/61.6 + (105-90.2)²/90.2 + (100-114.8)²/114.8 + (65-81.4)²/81.4 + (120-103.6)²/103.6 = 10.3311622159

Our degrees of freedom equates to (r-1)(c-1) where r is the number of rows and c is the number of columns. This means our degrees of freedom is df=(3-1)(2-1)=(2)(1)=2

Now we will find the p-value of our χ² test statistic by calculating χ²cdf(10.3311622159,1e99,2) which equals 0.0057097439 which is our p-value

Assuming a 5% significance level, since 0.0057<0.05, we reject the null hypothesis and conclude that the data doesn't provide evidence that the response to the television show is different for boys and girls. Since we found H: Observed=Expected to be true, that means it's more likely that there's not much of a difference between the observed and expected values.

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step-by-step explanation: