A random sample of 100 inmates at a maximum security prison...

A random sample of 100 inmates at a maximum security prison shows that exactly 10 of the respondents had been the victims of violent crime during their incarceration. Estimate the proportion of victims for the population as a whole, using the $90 \%$ confidence level. (HINT: Calculate the sample proportion $P_{s}$ before using Formula 7.3. Remember that proportions are equal to frequency divided by $N$.)

Key Concepts

Sample Proportion

The sample proportion is calculated by dividing the number of occurrences of an event by the total sample size. It represents the observed frequency of an attribute within the sample and serves as an estimate for the population proportion.

Confidence Interval for Proportions

A confidence interval for proportions provides a range of values within which the true population proportion is likely to fall, with a specified level of confidence. It is typically computed using the sample proportion, its standard error, and a critical value from the normal (z) distribution.

Confidence Level

The confidence level indicates the degree of certainty that the interval computed from the sample data contains the true population proportion. For example, a 90% confidence level implies that if the sampling process were repeated many times, approximately 90% of the computed intervals would include the true proportion.

Standard Error

The standard error measures the variability of the sample proportion from sample to sample. It is calculated as the square root of [p(1 - p) / n], where 'p' is the sample proportion and 'n' is the sample size, reflecting the precision of the estimated proportion.

Z-score

The z-score, or critical value from the standard normal distribution, corresponds to the desired confidence level. It is used to determine the margin of error in the confidence interval by multiplying it with the standard error.

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