Question

P= the proportion of Americans with Type A blood. p is unknown. Someone claims that " 30% of Americans have Type A blood". You suspect that the true proportion in the population is less than 30% and will do a hypothesis test. First you estimate p by randomly selecting 80 Americans, and 20 have Type A blood.

I. First, compute a 90% confidence interval for p.
II. What is the null (H 0 ) and alternative (H a) hypothesis for your hypothesis test?
III. In your sample, 20 out of 80 Americans had Type A blood. If instead you had observed 200 out of 800 , then the p-value of the hypothesis test would have been Circle one answer.

a) smaller
b) larger
c) the same
d) Not enough information has been provided to answer this question.

Answer 1

• I. The confidence interval falls between 0.166 and 0.334.
• II. The null hypothesis (H0) is p ≥ 0.3 and the alternative hypothesis (Ha) is p < 0.3.
• III. If the sample had observed 200 out of 800 Americans with Type A blood, the p-value of the hypothesis test would have been smaller.The answer is option ⇒A

Step-by-step explanation:

I.I. To compute a 90% confidence interval for p, we can use the sample proportion (20/80) to estimate the true proportion of Americans with Type A blood.

The formula to calculate the confidence interval for a proportion is:

Sample Proportion ± (Z * √(Sample Proportion * (1 - Sample Proportion) / Sample Size))

In this case, the sample proportion is 20/80 = 0.25, and the sample size is 80.

Using a standard normal distribution table or calculator, the critical value for a 90% confidence level is approximately 1.645.

Plugging in the values, we can calculate the confidence interval:

0.25 ± (1.645 * √(0.25 * (1 - 0.25) / 80))

The confidence interval is therefore (0.166, 0.334), meaning that we can be 90% confident that the true proportion of Americans with Type A blood falls between 0.166 and 0.334.

II. The null hypothesis (H0) for this hypothesis test would be:

H0: p ≥ 0.3

The alternative hypothesis (Ha) would be:

Ha: p < 0.3.

III. To determine whether the p-value of the hypothesis test would be smaller, larger, the same, or if not enough information has been provided, we need to consider the relationship between the sample size and the p-value.

In hypothesis testing, a larger sample size generally leads to a smaller p-value because a larger sample provides more evidence to either support or reject the null hypothesis. Therefore, if the sample size increased from 80 to 800, the p-value of the hypothesis test would likely be smaller.

So, the correct answer is a) smaller.