Question

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 is less 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

See the explanation below.

Explanation:

n = sample size = 1000

X = proportion of adults that a will = 450 / 1,000 = 0.45

P-value = 0.5

a = level of significance = 5%, or 0.05

A. At the 5% significance level, can you conclude that the percentage of people who have a will is less than 50%?

Our hypotheses are:

Null hypothesis: u ≥ 0.5

Alternate hypothesis: u < 0.5

Since this indicates a left tailed test, the decision rule is to reject null hypothesis if p-value is less than the level of significance.

`Z = test statistic = (0.45 - 0.5)/(0.5/√(1,000) )`

Z = - 0.05 / 0.0158113883008419 = - 3.1623

Using the excel function NORMSDIST(-3.1623), we have:

P- value = 0.000782701129001276, approximately 0.0008

Therefore, null hypothesis is rejected at 5% level of significance since the p-value 0.0008 is less than 5%, or 0.05.

Therefore, there is an enough reason to reject null hypothesis.

B. What is the Type I error in part A? What is the probability of making this error?

Since the probability is greater than 50%, the Type I error is therefore the statement in the question that less than 50% of adults have a will.

Also, the probability of making this Type I error is the 5% level of significance.

C. What would your decision be in part A if the probability of making a Type I error were zero?

The null hypothesis will be rejected if the probability of making a Type I error were zero in part A.