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Sample proportions serve as estimates of actual population proportions. In the context of the exercise, we calculate the sample proportion for males and females who preferred the Olympic gold medal. For males, the sample proportion, denoted as \( p_1 \), is computed as the number of male students selecting this option divided by the total number of male respondents. This yields a proportion of approximately 0.5648. Similarly, for females, the sample proportion \( p_2 \) is approximately 0.4319.

These proportions are vital because they form the basis for subsequent statistical calculations. They act as a snapshot of the sampled data that helps draw inferences about the broader population.

• Males: \( p_1 = \frac{109}{193} \approx 0.5648 \)
• Females: \( p_2 = \frac{73}{169} \approx 0.4319 \)

Understanding sample proportions helps in analyzing how representative the sample is of the total population and in setting up further calculations.

The standard error is a measure that reflects the variability of the sample statistic, providing insight into how much this statistic might fluctuate from sample to sample. In this example, we calculate the standard error for the difference between two sample proportions, which allows us to understand the precision of our estimate.

The formula used is:\[SE = \sqrt{ \frac{p_1(1 - p_1)}{n_1} + \frac{p_2(1 - p_2)}{n_2} }\]where \( n_1 \) and \( n_2 \) are the sizes of the male and female samples, respectively. After inserting the specific values, the calculated standard error \( SE \) is approximately 0.0561.

• This calculation is fundamental because it underpins the construction of confidence intervals.
• It indicates that the larger the sample size, the smaller the standard error, resulting in a more accurate estimation.

The z-score in statistics is a measure of how many standard deviations an element is from the mean. For confidence intervals, it helps in determining how much confidence we might have in our sample's estimate reflecting the true population.

For a 90% confidence interval, which is considered relatively precise, the corresponding z-score is approximately 1.645.

• This z-score is employed because it represents a balance between a higher confidence that requires a wider interval and the need for practical specificity.
• The z-score is integral to calculating the margin of error, as it directly impacts the size of the confidence interval.

Margin of Error (MOE) provides a range above and below the reported statistic that indicates where the true value is likely to lie, given a specific level of confidence. It is calculated by multiplying the standard error by the z-score:

\[MOE = SE \times z\]In this case, \( 0.0561 \times 1.645 \) gives us an MOE of approximately 0.0922.

• This MOE signifies that the true difference in preferences between male and female students is likely to lie within this range of the calculated difference.
• A lower MOE is preferred as it shows a narrow interval, indicating higher precision of our sample results.

Statistics education illuminates the power of understanding how to make inferences about populations from sample data. Through exercises like this, learners appreciate the real-world application of statistical concepts like confidence intervals.

Such exercises teach students critical skills:

• Understanding how sample data can provide insights into broader population behaviors.
• The process of constructing confidence intervals and interpreting their meaning.
• The importance of having accurate estimates and the reliability of these estimates when making data-driven decisions.

Through carefully structured statistical exercises, students enhance their analytical skills, becoming adept at using data for informed decision-making.