Statistics Quiz with Solutions (5 Questions)

To find \( P(-2 < Z < 2) \) for a standard normal variable \( Z \), we use the standard normal distribution table (z-table) or a standard normal distribution calculator.

The cumulative probability \( P(Z < 2) \) for a standard normal variable is approximately 0.9772.

The cumulative probability \( P(Z < -2) \) for a standard normal variable is approximately 0.0228.

Therefore, the probability that \( Z \) falls between -2 and 2 is:

\[
P(-2 < Z < 2) = P(Z < 2) - P(Z < -2) = 0.9772 - 0.0228 = 0.9544
\]

So, \( P(-2 < Z < 2) \) is approximately 0.9544 or 95.44%.

Descriptive statistics is the branch of statistics that involves organizing, displaying, and describing data.

To calculate the standard deviation of the population given the sample mean, sample size, and the standard deviation of the sample mean, we can use the relationship between the sample standard deviation (also known as the standard error of the mean) and the population standard deviation.

The formula for the standard error of the mean (\(\sigma_{\bar{x}}\)) is:

\[
\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}
\]

where \(\sigma\) is the population standard deviation and \(n\) is the sample size.

We are given:
- \(\sigma_{\bar{x}} = 20\)
- \(n = 400\)

We need to find \(\sigma\). Rearranging the formula to solve for \(\sigma\), we get:

\[
\sigma = \sigma_{\bar{x}} \times \sqrt{n}
\]

Substituting the given values:

\[
\sigma = 20 \times \sqrt{400}
\]

Since \(\sqrt{400} = 20\), we have:

\[
\sigma = 20 \times 20 = 400
\]

Therefore, the standard deviation of the population is 400.

In this context, the statistic is the proportion of people in the sample who are above the age of 80. The researcher calculated this proportion as follows:

\[
\hat{p} = \frac{120}{2400} = 0.05
\]

So, the value of the statistic is 0.05, or 5%. This represents the sample proportion of people above the age of 80 in the researcher's sample.

Qualitative data are measurements for which there is no natural numerical scale, but which consist of attributes, labels, or other nonnumerical characteristics.