Suppose a new treatment for a certain disease is given to a sample of n=200 patients. The treatment was successful for 166 of the patients. Assume these patients are representative of the population of individuals who have this disease. (a) Calculate the sample proportion of patients who were successfully treated. (b) Calculate the 95% confidence interval for the population proportion of patients who are successfully treated. (c) Write a sentence that correctly interprets this interval. (3.0 points)

CI95= 0.855 or 0.805

Why 1.96 in the formulae ?

(a) To calculate the sample proportion of patients who were successfully treated, divide the number of patients successfully treated (166) by the total sample size (n=200):

Sample Proportion = Number of patients successfully treated / Total sample size

Sample Proportion = 166 / 200 = 0.83

The sample proportion of patients who were successfully treated is 0.83.

(b) To calculate the 95% confidence interval for the population proportion of patients who are successfully treated, we can use the formula:

Confidence Interval = Sample Proportion ± (Critical Value) * (Standard Error)

First, we need to find the standard error. The standard error can be calculated using the formula:

Standard Error = √((Sample Proportion * (1 - Sample Proportion)) / Sample Size)

Standard Error = √((0.83 * (1 - 0.83)) / 200) ≈ 0.0245

Next, we need to find the critical value corresponding to the desired confidence level. For a 95% confidence level, the critical value is approximately 1.96.

Confidence Interval = 0.83 ± 1.96 * 0.0245

Confidence Interval = 0.83 ± 0.048

Thus, the 95% confidence interval for the population proportion of patients who are successfully treated is approximately (0.782, 0.878).

(c) A sentence that correctly interprets this interval would be: "We are 95% confident that the true population proportion of patients who are successfully treated lies between 0.782 and 0.878."

(a) To calculate the sample proportion of patients who were successfully treated, you can divide the number of patients who were successfully treated (166) by the total number of patients in the sample (200).

Sample Proportion = Number of successful treatments / Total sample size
Sample Proportion = 166 / 200
Sample Proportion = 0.83 or 83%

The sample proportion of patients who were successfully treated is 0.83 or 83%.

(b) To calculate the 95% confidence interval for the population proportion, you can use the formula:

Confidence Interval = Sample Proportion ± Margin of Error

The margin of error depends on the desired confidence level, sample proportion, and sample size. For a 95% confidence level and a large enough sample size (n = 200 is generally considered large enough), the standard formula can be used:

Margin of Error = 1.96 * √(Sample Proportion * (1 - Sample Proportion) / Sample Size)

Margin of Error = 1.96 * √(0.83 * (1 - 0.83) / 200)

Calculating this, we get:

Margin of Error ≈ 0.05 or 5% (rounded to two decimal places)

Therefore, the 95% confidence interval for the population proportion of patients who are successfully treated is:

Confidence Interval = Sample Proportion ± Margin of Error
Confidence Interval = 0.83 ± 0.05
Confidence Interval = (0.78, 0.88) or 78% to 88%

(c) A sentence that correctly interprets this interval could be: "We are 95% confident that the true proportion of patients who are successfully treated with this new treatment lies between 78% and 88%."

The following is a hint for you to get started.

Use a confidence interval formula for proportions:

CI95 = p ± (1.96)(√pq/n)
...where p = x/n, q = 1 - p, and n = sample size.

Using your data:
p = 166/200
q = 1 - p
n = 200

Convert fractions to decimals. Calculate using the formula above.

I'll let you take it from here.