Question

1. lets focus now more closely on the distribution of sample proportions from samples of size ( n = 75 ). we can see from the table on the previous page that this distribution is approximately normal in shape, with a mean of 0.41 and a standard deviation of 0.0568. given the normal shape of the distribution of sample proportions, we can apply the empirical rule. use what you know about that rule to fill in the blanks below.

a. approximately 68% of the sample proportions from samples of size ( n = 75 ) are between ______________ and ______________.
b. approximately 95% of the sample proportions from samples of size ( n = 75 ) are between ______________ and ______________.

Explanation:

Step1: Recall the Empirical Rule

The Empirical Rule states that for a normal distribution:

• Approximately \(68\%\) of the data lies within \(1\) standard deviation (\(\sigma\)) of the mean (\(\mu\)).
• Approximately \(95\%\) of the data lies within \(2\) standard deviations (\(\sigma\)) of the mean (\(\mu\)).
• Approximately \(99.7\%\) of the data lies within \(3\) standard deviations (\(\sigma\)) of the mean (\(\mu\)).

Given \(\mu = 0.41\) and \(\sigma=0.0568\)

Step2: Calculate for \(68\%\)

For the \(68\%\) interval:
Lower bound \(=\mu - \sigma=0.41 - 0.0568 = 0.3532\)
Upper bound \(=\mu+\sigma = 0.41+ 0.0568=0.4668\)

Step3: Calculate for \(95\%\)

For the \(95\%\) interval:
Lower bound \(=\mu - 2\sigma=0.41-2\times0.0568=0.41 - 0.1136 = 0.2964\)
Upper bound \(=\mu + 2\sigma=0.41+2\times0.0568=0.41 + 0.1136=0.5236\)

Answer:

A. Approximately \(68\%\) of the sample proportions from samples of size \(n = 75\) are between \(0.3532\) and \(0.4668\).
B. Approximately \(95\%\) of the sample proportions from samples of size \(n = 75\) are between \(0.2964\) and \(0.5236\).