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The sample proportion is a statistical tool used to make educated guesses about a larger population based on a smaller sample from that same population. Imagine you have a jar of candies, and you randomly pick a few to check how many are red. Here, the red candies in your hand represent the sample, and you're interested in estimating their proportion in the entire jar.In the exercise above, calculating the sample proportion involves dividing the number of manual transmission cars by the total number of cars sampled. This calculation helps us understand what fraction of college students in the study drive manual cars. A proportion in this context is represented by the symbol \( \hat{p} \), signifying the ratio of interest. For the exercise, you get \( \hat{p} = \frac{26}{122} \approx 0.213 \). This means approximately 21.3% of the sample drives a manual transmission, giving us a snapshot to make further evaluations about the entire college student population.

The standard error of proportion measures the variability or spread of the sample proportion in repeated sampling. It works much like a "margin of error", telling us how much we can expect the sample proportion to differ from the true population proportion due to random sampling.To calculate this, use the formula: \[ SE = \sqrt{ \frac{\hat{p}(1-\hat{p})}{n} } \]where:\( \hat{p} \) is the sample proportion, and \( n \) is the sample size. For our scenario with manual transmissions, the calculation became:\[ SE = \sqrt{ \frac{0.213(1-0.213)}{122} } \approx 0.038 \]This value, 0.038, shows the typical deviation of our sample proportion from the true proportion if the experiment were repeated with new samples of the same size. A smaller SE indicates a more precise estimate of the population proportion.

The Z-score is a critical value used in statistics that corresponds to a specified confidence level. It works as a multiplier of the standard error to stretch or shrink the confidence interval around the sample proportion. Choosing the appropriate Z-score is a key step in building accurate confidence intervals. For a given confidence level, look up a Z-score that aligns with it from a Z-table, which relates to how many standard deviations away from the mean we need to consider. A 90% confidence level is associated with a Z-score of approximately 1.645. This choice accounts for 90% of the data under the normal distribution curve, leaving 5% in each tail. By multiplying the standard error by this Z-score, we extend the interval to cover the range within which we predict the true proportion lies with the desired confidence.

Statistical analysis allows us to glean insights and determine truths from data. In this exercise, we're able to use sample data to make claims about a larger population, quantifying uncertainty and making informed conclusions.By calculating a confidence interval using the sample proportion and standard error, we're performing statistical analysis. The formula used:\[ \hat{p} \pm Z \times SE \]is critical for telling us how much we can rely on our sample to predict the behavior of the larger group. For our college students' manual transmission estimate, you find:- Lower bound: \(0.213 - 1.645 \times 0.038 \approx 0.150\)- Upper bound: \(0.213 + 1.645 \times 0.038 \approx 0.276\)Thus, a 90% confidence interval \((0.150, 0.276)\) indicates that we estimate with 90% certainty that the true proportion of manual transmission users lies within this range.These statistical methods translate sample observations into powerful predictions about populations, helping us make decisions backed by data.