A physical therapist wants to determine the difference in the proportion of men and women who participate in regular sustained physical activity. What sample size should be obtained if he wishes the estimate to be within four percentage points with 99

​% confidence, assuming that
​(a) he
uses the estimates of 22.8​% male and 19.9 % female from a previous​ year?
​(b) he does not use any prior​ estimates?

To calculate the sample size needed, we can use the formula for the sample size for estimating proportions:

n = (Z^2 * p * q) / E^2

where:
n = sample size
Z = z-score for the desired level of confidence
p = estimated proportion of the population
q = 1 - p
E = maximum allowable error (in this case, 4 percentage points)

(a) If he uses the estimates of 22.8% male and 19.9% female from the previous year:
Assuming he wants to estimate the difference between proportions, we need to calculate the sample size separately for men and women.

For men:
n1 = (Z^2 * p1 * q1) / E^2
= (Z^2 * 0.228 * (1 - 0.228)) / (0.04^2)

For women:
n2 = (Z^2 * p2 * q2) / E^2
= (Z^2 * 0.199 * (1 - 0.199)) / (0.04^2)

Total sample size:
n = n1 + n2

(b) If he does not use any prior estimates:
In this case, we assume a conservative estimate of 0.5 for p (since it gives the maximum required sample size).

n = (Z^2 * p * q) / E^2
= (Z^2 * 0.5 * 0.5) / (0.04^2)

Now, we need to determine the value of Z for the desired level of confidence.

For a 99% confidence level, the Z-score is approximately 2.576.

Substitute the appropriate values into the formulas to calculate the sample sizes.

Note: The calculations provided here are based on basic statistical formulas. It's important to consult with a statistician or use specialized software for accurate and detailed analysis.

To determine the sample size needed, we can use the formula for sample size in a proportion estimation:

n = (Z^2 * p * (1 - p)) / E^2

Where:
- n is the sample size
- Z is the Z-score corresponding to the desired confidence level (99% in this case)
- p is the estimated proportion
- E is the desired margin of error

Let's calculate the sample size in both scenarios:

(a) Using the estimates from a previous year:
Given estimates of 22.8% male (p1) and 19.9% female (p2), we can calculate the sample sizes for each proportion first.

n1 = (Z^2 * p1 * (1 - p1)) / E^2
n2 = (Z^2 * p2 * (1 - p2)) / E^2

Since we want to compare proportions, we use the larger sample size from the two estimates.

n = max(n1, n2)

(b) Without using any prior estimates:
In this case, we assume a 50% proportion for both men and women because that would give us the largest sample size. So, p = 0.5 in the formula.

Now, let's plug in the values and calculate the sample sizes.

(a) Using the estimates from a previous year:
Assuming a desired margin of error (E) of 4 percentage points and a 99% confidence level (which corresponds to a Z-score of approximately 2.58), we have:

n1 = (2.58^2 * 0.228 * (1 - 0.228)) / (0.04^2)
n2 = (2.58^2 * 0.199 * (1 - 0.199)) / (0.04^2)

n1 ≈ 437
n2 ≈ 392

Thus, the sample size required would be the larger of the two: n ≈ 437.

(b) Without using any prior estimates:
Assuming a desired margin of error (E) of 4 percentage points, a 99% confidence level (which corresponds to a Z-score of approximately 2.58), and p = 0.5, we have:

n = (2.58^2 * 0.5 * (1 - 0.5)) / (0.04^2)
n ≈ 1067

Therefore, the sample size needed when not using any prior estimates would be approximately 1067.

Note: The sample sizes calculated using these formulas provide a rough estimate. In practice, it's always a good idea to consult with a statistician to ensure accurate sample size determination.

A physical therapist wants to determine the difference in the proportion of men and women who participate in regular sustained physical activity. What sample size should be obtained if hehe wishes the estimate to be within fourfour percentage points with 9595​% ​confidence, assuming that

​(a) hehe uses the estimates of 21.821.8​% male and 18.518.5​% female from a previous​ year?
​(b) hehe does not use any prior​ estimates?

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