Question

In a recent poll of 750750 randomly selected​ adults, 589589 said that it is morally wrong to not report all income on tax returns. Use a 0.050.05 significance level to test the claim that 7575​% of adults say that it is morally wrong to not report all income on tax returns. Identify the null​ hypothesis, alternative​ hypothesis, test​ statistic, P-value, conclusion about the null​ hypothesis, and final conclusion that addresses the original claim. Use the​ P-value method. Use the normal distribution as an approximation of the binomial distribution.

Answer 1

We conclude that % of adults who say that it is morally wrong to not report all income on tax returns is different from 75%.

Explanation:

We are given that in a recent poll of 750 randomly selected​ adults, 589 said that it is morally wrong to not report all income on tax returns.

We have to test the claim that 75% of adults say that it is morally wrong to not report all income on tax returns.

Let p = % of adults who say that it is morally wrong to not report all income on tax returns.

So, Null Hypothesis,
`H_0` : p = 75% {means that % of adults who say that it is morally wrong to not report all income on tax returns is 75%}

Alternate Hypothesis,
`H_0` : p
`\\eq` 20% {means that % of adults who say that it is morally wrong to not report all income on tax returns is different from 75%}

The test statistics that would be used here One-sample z proportion statistics;

T.S. =
`\frac{\hat p-p}{\sqrt{(\hat p(1-\hat p))/(n) } }` ~ N(0,1)

where,
`\hat p` = sample proportion of adults who said that it is morally wrong to not report all income on tax returns =
`(589)/(750)` = 0.785

n = sample of adults = 750

So, test statistics =
`\frac{0.785-0.75}{\sqrt{(0.785(1-0.785))/(750) } }`

= 2.33

The value of z test statistics is 2.33.

Now, P-value of the test statistics is given by the following formula;

P-value = P(Z > 2.33) = 1 - P(Z
`\leq` 2.33)

= 1 - 0.9901 = 0.0099

Also, P-value for two-tailed test is calculated as = 2
`*` 0.0099 = 0.0198

Since, the P-value of test statistics is less than the level of significance as 0.0198 < 0.05, so we have sufficient evidence to reject our null hypothesis as it will fall in the rejection region due to which we reject our null hypothesis.

Therefore, we conclude that % of adults who say that it is morally wrong to not report all income on tax returns is different from 75%.