Question

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 \(N\).)

Short answer

The 90% confidence interval for the proportion of victims is (0.05135, 0.14865).

Step-by-step solution

Calculate the Sample Proportion

The sample proportion of inmates who have been victims of violent crime is given by the formula:\[P_s = \frac{X}{N}\]where \(X\) is the number of inmates who reported being victims (10), and \(N\) is the total number of inmates in the sample (100). Hence,\[P_s = \frac{10}{100} = 0.1\]

Identify the Standard Error

The standard error (SE) of the sample proportion is calculated using the formula:\[SE = \sqrt{\frac{P_s (1 - P_s)}{N}}\]Substitute the values obtained:\[SE = \sqrt{\frac{0.1 (1 - 0.1)}{100}} = \sqrt{\frac{0.1 \times 0.9}{100}} = \sqrt{0.0009} = 0.03\]

Find the Z-value for 90% Confidence Level

Using the Z-table, the Z-value corresponding to a 90% confidence level is 1.645.

Calculate the Confidence Interval

The confidence interval (CI) can be calculated using the formula:\[CI = P_s \pm Z \times SE\]Substituting the values:\[CI = 0.1 \pm 1.645 \times 0.03\]This results in:\[Lower \; Limit = 0.1 - 1.645 \times 0.03 = 0.05135\]\[Upper \; Limit = 0.1 + 1.645 \times 0.03 = 0.14865\]Thus, the 90% confidence interval is \(0.05135, 0.14865\).

Key concepts

sample proportion

In statistics, the sample proportion is a simple yet crucial concept used when working with surveys or samples. It represents the percentage of a certain trait within a sample group.

For instance, in the given exercise, we wanted to find out how many inmates in the sample had been victims of violent crime. By dividing the number of victims (10) by the total number of inmates surveyed (100), we calculated the sample proportion: \[P_s = \frac{10}{100} = 0.1\]This proportion informs us that 10% of the sampled inmates experienced violent crime, providing a snapshot into the larger population's dynamics.

Understanding the sample proportion is fundamental since it is the base for further calculations, such as standard error and confidence intervals.

standard error

The standard error (SE) quantifies the uncertainty or variability of the sample proportion. It plays a vital role in inferential statistics, helping us estimate how much the sample proportion deviates from the true population proportion.

The formula for calculating the standard error of a sample proportion, as shown in the exercise, is:\[SE = \sqrt{\frac{P_s (1 - P_s)}{N}}\]Here, \(P_s\) is the sample proportion,\(1 - P_s\) accounts for the proportion of the sample that doesn't have the characteristic, and \(N\) is the total sample size.

Using our values, we get:\[SE = \sqrt{\frac{0.1 (1 - 0.1)}{100}} = \sqrt{\frac{0.1 \times 0.9}{100}} = \sqrt{0.0009} = 0.03\]This value tells us how much the sample proportion 0.1 might vary if we took different samples of the same size from the same population.

Z-value

The Z-value, or Z-score, is a statistical measure that indicates how many standard deviations an element is from the mean. In the context of confidence intervals, it represents the number of standard errors that correspond to a certain confidence level.

Confidence levels show how confident we are that the true population parameter lies within our calculated interval. Common confidence levels are 90%, 95%, and 99%. Each has an associated Z-value which can be found in statistical Z-tables.

For example, a 90% confidence level corresponds to a Z-value of 1.645. This means that to capture the true population proportion within 90% certainty, we need to account for 1.645 standard errors in either direction of our sample proportion value.

confidence level

A confidence level is a measure that indicates how often the true parameter of a population would be captured by the interval estimate, if we were to take many samples and build intervals for each one.

In simpler terms, a 90% confidence level means that we would expect the true proportion to lie within the interval 90 times out of 100 samples. The process of creating a confidence interval involves using the Z-value and standard error to construct a range around the sample proportion.For instance, using the formula:\[CI = P_s \pm Z \times SE\]With \(P_s = 0.1,\) \(Z = 1.645,\) \(SE = 0.03,\)We get:\[CI = 0.1 \pm 1.645 \times 0.03 = [0.05135, 0.14865]\]This interval suggests that with 90% confidence, the true proportion of inmates who are victims of violent crime lies between 5.135% and 14.865%.