Question

Hypertension is when an adult is classified as having high blood pressure (above 130 systolic blood pressure is considered hypertension). Researchers want to know the proportion of adult North Americans (above age of 18) that have hypertension. Based on a study of 3532 adult North Americans, 1219 of them were classified as having hypertension.

a. Researchers want to test if more than a quarter of all North American adults have hypertension (that is to say more than 25% proportion of North American adults). State the null and alternative hypothesis in proper notation.
b. Create a 95% confidence interval for the true proportion of adult North Americans that have hypertension. Interpret this interval in context of the study.c. Say your 95% confidence interval is (0.329 , 0.361). Can you say with a high degree of confidence that more than a quarter of all North Americans have hypertension. Explain in a sentence or two.
d. If we were to decrease our level of confidence, what would we expect to happen to the confidence interval? Get wider/ get narrower/ stay the same ?

Answer 1

a) The null and alternative hypotheses are
`H_0`: p≤ 0.25

`H_1` : p 0.25

b) 95% Confidence interval is (0.329, 0.361)

c) Both the limits of the confidence interval are greater than 0.25

d) If we were to decrease our level of confidence, lower. happen to the confidence interval

The provided values are n = 3,532

X = 1,219

Also, p = X 1,219 3,532 0.3451

a) The null and alternative hypotheses are

`H_0`: p≤ 0.25

`H_1`: p > 0.25

b) 95% confidence interval for the true proportion of adult North Americans that have hypertension

1219/3532 = 0.35

Confidence interval = sample proportion ± margin of error

Margin of error = z × √pq/n

p = 0.35

q = 1 - 0.35 = 0.65

z score for 95% confidence level is 1.96

Margin of error = 1.96 × √(0.35 × 0.65)/3532 = 0.016

Confidence interval = 0.35 ± 0.016

Confidence Interval = (0.329 , 0.361)

c) The probability that the position of a statistical parameter in a sample survey is accurate for the population is indicated by the confidence level.

The interval of 95% confidence is (0.329, 0.361).

The confidence interval's two limits are both larger than 0.25.

It can be said with great confidence that over 25% of North Americans suffer from hypertension.

d) The confidence interval set is narrower if the confidence level is lowered since the confidence interval's breadth rises with confidence level.

Answer 2

Explanation:

a) We would set up the hypothesis test.

For the null hypothesis,

H0 : p ≥ 0.25

For the alternative hypothesis,

H1 : p < 0.25

b) from the given information,the sample proportion or point estimate for the population proportion is

1219/3532 = 0.35

Confidence interval = sample proportion ± margin of error

Margin of error = z × √pq/n

p = 0.35

q = 1 - 0.35 = 0.65

z score for 95% confidence level is 1.96

Margin of error = 1.96 × √(0.35 × 0.65)/3532 = 0.016

Confidence interval = 0.35 ± 0.016

c) Given that the 95% confidence interval is (0.329 , 0.361), it means that the lower limit of the confidence interval is 0.329 and the upper limit is 0.361

If more than a quarter of all North Americans have hypertension. It means that the true proportion can be within this interval. 95% confidence interval is a high degree of confidence. Therefore, we can say that with a high degree of confidence that more than a quarter of all North Americans have hypertension.

d) it would get narrower