Statistics Quiz with Solutions (5 Questions)

A random variable is called **continuous** if its possible values contain a whole interval of numbers.

Discrete random variables arise from a counting process.

Given the dataset \(D = \{1.5, 2.3, 1.7, 3.6, 2.9, 4.2\}\), we first sort it to get \(D_{\text{sorted}} = \{1.5, 1.7, 2.3, 2.9, 3.6, 4.2\}\).

To find the second quartile (Q2), which is the median:

- Since the number of observations in \(D_{\text{sorted}}\) is even (\(n = 6\)), Q2 is the average of the middle two values. Thus,
\[ Q2 = \frac{D_{\text{sorted}}[3] + D_{\text{sorted}}[4]}{2} \]
\[ Q2 = \frac{2.3 + 2.9}{2} \]
\[ Q2 = \frac{5.2}{2} \]
\[ Q2 = 2.6 \]

Therefore, the second quartile (Q2) of the dataset is \(2.6\).

To calculate the sample mean (\(\bar{x}\)) of the cholesterol levels for the selected sample:

Given the cholesterol levels of the 7 patients: \(128\), \(127\), \(153\), \(144\), \(132\), \(120\), \(115\),

The formula for the sample mean is:

\[ \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} \]

Where:
- \(x_i\) represents each value in the dataset,
- \(n\) is the number of observations in the dataset.

Substituting the given values into the formula:

\[ \bar{x} = \frac{128 + 127 + 153 + 144 + 132 + 120 + 115}{7} \]

\[ \bar{x} = \frac{919}{7} \]

\[ \bar{x} \approx 131.29 \]

Therefore, the sample mean of the cholesterol levels for the selected sample of 7 patients is approximately \(131.29\) mg/dL.

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

Given that the population standard deviation (\( \sigma \)) is \(2\) inches and the sample size (\( n \)) is \(100\), the calculation is:

\[ \sigma_{\bar{x}} = \frac{2}{\sqrt{100}} \]

\[ \sigma_{\bar{x}} = \frac{2}{10} \]

\[ \sigma_{\bar{x}} = 0.2 \]

Therefore, the standard deviation of the sample mean for this random sample of 100 male students is \(0.2\) inches.