Manual Transmission Automobiles In 2014, six percent of the...

Manual Transmission Automobiles In $2014,$ six percent of the cars sold had a manual transmission. A random sample of college students who owned cars revealed the following: out of 122 cars, 26 had manual transmissions. Estimate the proportion of college students who drive cars with manual transmissions with $90 \%$ confidence.

Manual Transmission Automobiles In $2014,$
six percent of the cars sold had a manual transmission.
A random sample of college students who owned cars
revealed the following: out of 122 cars, 26 had
manual transmissions. Estimate the proportion of
college students who drive cars with manual
transmissions with $90 \%$ confidence.

Key Concepts

Point Estimation of a Proportion

This concept involves using a sample to estimate the true proportion of a population possessing a certain characteristic. By calculating the sample proportion (the number of successes divided by the total sample size), we obtain a point estimate that serves as our best guess for the population proportion.

Confidence Interval

A confidence interval provides a range of values, derived from the sample statistic, that is likely to contain the true population parameter. In the context of proportions, it is constructed around the sample proportion and reflects the degree of uncertainty associated with the estimate, typically expressed with a confidence level such as 90% or 95%.

Standard Error of a Proportion

The standard error quantifies the variability of the sample proportion due to random sampling. It is calculated by taking the square root of [p(1-p)/n], where p is the sample proportion and n is the sample size. This measure is crucial for determining the precision of the point estimate and for constructing the confidence interval.

Z-Score (Critical Value) for Confidence Level

For large samples, the sampling distribution of the sample proportion approximates a normal distribution, allowing us to use z-scores to define the confidence interval. The z-score corresponding to a given confidence level determines the multiplier used on the standard error to set the width of the interval.

Central Limit Theorem for Proportions

The central limit theorem states that, for sufficiently large sample sizes, the distribution of the sample proportion will be approximately normal regardless of the distribution of the population. This approximation justifies the use of z-scores in constructing confidence intervals for proportions.

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