Questions: Formulas The normal approximation will also give you values of the standardized statistic and p-value based on its mathematical predictions. As you learned previously, the standardized score is calculated as standardized statistic = z = (statistic - mean of null distribution) / (standard deviation of null distribution) The mean of the null distribution is the hypothesized value of the long-run proportion (π). The standard deviation can be obtained in two ways: First, find the standard deviation of the null distribution by simulating. Second, predict the value of the standard deviation by substituting into this formula: sqrt((π(1-π))/n) PARTICIPATION ACTivity 2.10.14: Question 8. Use the formula to determine the (theoretical; predicted) standard deviation of the sample proportion. Then compare this to the SD from your simulated sample proportions, as recorded in #5a. Are they similar?

Solution Steps

The theoretical standard deviation of the sample proportion is calculated using the formula:

\[ \sigma = \sqrt{\frac{\pi(1 - \pi)}{n}} \]

Substituting the values \( \pi = 0.5 \) and \( n = 100 \):

\[ \sigma = \sqrt{\frac{0.5(1 - 0.5)}{100}} = \sqrt{\frac{0.5 \cdot 0.5}{100}} = \sqrt{\frac{0.25}{100}} = \sqrt{0.0025} = 0.0500 \]

Thus, the theoretical standard deviation is \( 0.0500 \).

The mean \( \mu \) of the simulated sample proportions is calculated as follows:

\[ \mu = \frac{\sum x_i}{n} = \frac{4.97}{10} = 0.497 \]

Next, the variance \( \sigma^2 \) is calculated using the formula for sample variance:

\[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{n - 1} = 0.0004 \]

The standard deviation is then:

\[ \sigma = \sqrt{0.0004} = 0.0189 \]

Thus, the simulated standard deviation is \( 0.0189 \).

To determine if the theoretical and simulated standard deviations are similar, we compare:

• Theoretical Standard Deviation: \( 0.0500 \)
• Simulated Standard Deviation: \( 0.0189 \)

The difference between the two values is significant, leading to the conclusion that they are not similar.

Final Answer