Question

9.Rao’s Finance claims that less than 50% of adults in Suva have a will. A will is a legal document that sets forth your wishes regarding the distribution of your property and the care of any minor children. If you die without a will, those wishes may not be carried out. To substantiate the claim, a random sample of 1000 adults showed that 450 of them have a will. A. At the 5% significance level, can you conclude that the percentage of people who have a will than 50%? B. What is the Type I error in part A? What is the probability of making this error? C. What would your decision be in part A if the probability of making a Type I error were zero? Explain

Answer 1

(A) We conclude that the percentage of people who have a will is less than 50% at 5% significance level.

(B) Type I error is that we conclude that the percentage of people who have a will is less than 50% but in actual it was equal to 50%.

Probability of making this error is 5%.

(C) If the probability of making a Type I error were zero in part A, then we will definitely be able to reject our null hypothesis.

Step-by-step explanation:

We are given that Rao’s Finance claims that less than 50% of adults in Suva have a will.

To substantiate the claim, a random sample of 1000 adults showed that 450 of them have a will.

Let p = percentage of people who have a will.

So, Null Hypothesis, $H_0$ : p = 50%      {means that the percentage of people who have a will is equal to 50%}

Alternate Hypothesis, $H_A$ : p < 50%     {means that the percentage of people who have a will is less than 50%}

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

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

where, $\hat p$ = sample proportion of people who have a will = $\frac{450}{1000}$ = 0.45

           n = sample of adults = 1000

So, the test statistics  =  $\frac{0.45-0.50}{\sqrt{\frac{0.50(1-0.50)}{1000} } }$  

                                     =  -3.162

The value of z test statistic is -3.162.

(A) Now, at 5% significance level the z table gives critical value of -1.645 for left-tailed test.

Since our test statistic is less than the critical value of z as -3.162 < -1.645, 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 the percentage of people who have a will is less than 50%.

(B) Type I error states that Probability of rejecting null hypothesis given the fact that it was true or in other words; Probability of rejecting a true hypothesis.

SO, according to our question Type I error is that we conclude that the percentage of people who have a will is less than 50% but in actual it was equal to 50%.

Probability of making this error = Level of significance = 5%

(C) If the probability of making a Type I error were zero in part A, then we will definitely be able to reject our null hypothesis and conclude that the percentage of people who have a will is less than 50%.