Question

According to the 2017-2018 U.S. Pet Ownership & Demographics Sourcebook, 38.4% of US households own a dog. Ms Ferguson believes the percent of households who own a dog has changed since 2018, so she polls 90 students and finds that 39 students own a dog. Test Ms. Ferguson’s claim at the α = 0.05 significance level. Regardless of which method you use, sketch a graph showing the critical region and where the test statistic falls - inside or outside the critical region.

Answer 1

Let the population proportion of US households who own a dog be P

Step 1

H0: P= 0.384 (counter claim)

H1: P ≠ 0.384 (claim)

This is a TWO-tailed one sample Z test.

Step 2

Write down the data you have,

The sample proportion (p) = 39/90 = 0.433

The population proportion (P) = 0.384

The standard deviation (σ) = sqrt [P * (1 - P) / n] = sqrt (0.384*0.616/90) = 0.05126

Number of observations (n) = 90

Step 3

Z score test statistic = z = (p - P) / σ = (0.433 - 0.384)/0.05126 = 0.956

critical region:

critical value = Z1 - alpha/2 = Z0.975 = 1.96  (as alpha =0.05)

so, the critical region or rejection region lies a z value greater than 1.96 or lesser than -1.96

we will reject the null hypothesis if we get a test statistic lying in these rejection regions.

graphically,

as we can see that the z statistics lies in the acceptance region so we fail to reject the null hypothesis.

or

Since we have a two-tailed test, the p-value is the probability that the z-score is less than -0.956 or greater than 0.956

P value = P (z < - 0.956) + P (z > 0.956) = 0.34 > 0.05 using standard normal table

Since it is greater than alpha = 0.05 hence, we fail to reject the null hypothesis.

Conclusion: That is there is no sufficient evidence to support the claim that current percent of US households who own a dog is different from the previous year percentage.