Question

A random sample of 1496 respondents of a major metropolitan area was questioned about a number of issues. Construct estimates to the population at the $95\mathrm{\%}$ level for each of the results reported next. Express the final confidence interval in percentages (e.g., "between 40 and $45\mathrm{\%}$ agreed that premarital sex was always wrong"). a. When asked to agree or disagree with the statement "Explicit sexual books and magazines lead to rape and other sex crimes," 823 agreed. b. When asked to agree or disagree with the statement "Hand guns should be outlawed," 650 agreed. c. 375 of the sample agreed that marijuana should be legalized. d. 1023 of the sample said that they had attended a church, synagogue, mosque, or other place of worship at least once within the past month. e. 800 agreed that public elementary schools should have sex education programs starting in the fifth grade.

Short answer

Between 52.5% and 56.5% agreed with statement a; between 41.2% and 45.6% agreed with statement b; between 23.3% and 26.7% agreed with statement c; between 66.1% and 70.7% agreed with statement d; between 51.7% and 55.7% agreed with statement e.

Step-by-step solution

- Determine the sample proportion

Calculate the sample proportion (p) for each statement. The sample proportion is the ratio of the number of respondents who agreed to the total number of respondents in the sample.For statement a: $p_a=\frac{823}{1496}$For statement b: $p_b=\frac{650}{1496}$For statement c: $p_c=\frac{375}{1496}$For statement d: $p_d=\frac{1023}{1496}$For statement e: $p_e=\frac{800}{1496}$

- Calculate the standard error

The standard error (SE) of the sample proportion is calculated using the formula: $SE=\sqrt{\frac{p(1-p)}{n}}$, where n is the sample size (1496). Calculate the standard error for each statement.For statement a: $S{E}_a=\sqrt{\frac{p_a(1-p_a)}{1496}}$For statement b: $S{E}_b=\sqrt{\frac{p_b(1-p_b)}{1496}}$For statement c: $S{E}_c=\sqrt{\frac{p_c(1-p_c)}{1496}}$For statement d: $S{E}_d=\sqrt{\frac{p_d(1-p_d)}{1496}}$For statement e: $S{E}_e=\sqrt{\frac{p_e(1-p_e)}{1496}}$

- Find the Z-score for the confidence level

For a 95% confidence level, the Z-score is approximately 1.96.

- Calculate the margin of error

The margin of error (MOE) is calculated using the formula: $MOE=Z\times SE$Calculate the margin of error for each statement using the Z-score (1.96) and the standard error.For statement a: $MO{E}_a=1.96\times S{E}_a$For statement b: $MO{E}_b=1.96\times S{E}_b$For statement c: $MO{E}_c=1.96\times S{E}_c$For statement d: $MO{E}_d=1.96\times S{E}_d$For statement e: $MO{E}_e=1.96\times S{E}_e$

- Construct the confidence intervals

The confidence interval (CI) is calculated using the formula: $CI=p\pm MOE$Convert the interval to a percentage and round to one decimal place if necessary.For statement a: Convert $p_a-MO{E}_a$ and $p_a+MO{E}_a$ to percentages.For statement b: Convert $p_b-MO{E}_b$ and $p_b+MO{E}_b$ to percentages.For statement c: Convert $p_c-MO{E}_c$ and $p_c+MO{E}_c$ to percentages.For statement d: Convert $p_d-MO{E}_d$ and $p_d+MO{E}_d$ to percentages.For statement e: Convert $p_e-MO{E}_e$ and $p_e+MO{E}_e$ to percentages.

Key concepts

Sample Proportion

The sample proportion, denoted as $p$, is a fundamental concept in statistics. It represents the fraction of members in the sample that exhibit a specific characteristic. For instance, if 823 out of 1496 respondents agree with the statement 'Explicit sexual books and magazines lead to rape and other sex crimes,' the sample proportion is calculated as follows: $p=\frac{823}{1496}$. This value helps us understand the behavior or opinion of the sample group relative to the entire population being studied. To express it as a percentage, simply multiply it by 100.

Standard Error

The Standard Error (SE) measures the variability or precision of the sample proportion. It tells us how much the sample proportion is expected to fluctuate across different random samples from the population. The formula for SE in terms of proportion is given by: $SE=\sqrt{\frac{p(1-p)}{n}}$ where $p$ is the sample proportion, and $n$ is the sample size. Using the provided example, assume $p=0.55$, and the sample size $n=1496$; $SE=\sqrt{\frac{0.55(1-0.55)}{1496}}$ This results in a specific value for SE, which is crucial in further calculating confidence intervals.

Margin of Error

The Margin of Error (MOE) provides a range within which we expect the true population proportion to lie. It accounts for random sampling errors and is directly dependent on the confidence level and standard error. MOE is calculated using the formula: $MOE=Z\times SE$, where $Z$ is the Z-score corresponding to the desired confidence level (usually 1.96 for a 95% confidence level). For example, if the SE is 0.012, then: $MOE=1.96\times 0.012=0.02352$, translating to 2.35% when expressed as a percentage. Hence, the margin of error helps convey how accurate our sample statistic is likely to be.

Z-score

The Z-score is a statistical measure that describes a value's relationship to the mean of a group of values. For confidence intervals, it determines how many standard deviations an element is from the mean. For typical confidence levels: - A 95% confidence level corresponds to a Z-score of approximately 1.96. - A 99% confidence level corresponds to a Z-score of approximately 2.58. These values are based on the standard normal distribution. Thus, for a 95% confidence interval, we use a Z-score of 1.96 to ensure our interval captures the expected proportion of the population with 95% certainty.

95% Confidence Level

A 95% confidence level means that if we were to take many samples and build a confidence interval from each one, we would expect approximately 95% of these intervals to contain the true population parameter. It's a way of indicating the reliability or certainty of the estimate. For example, if we calculate a confidence interval for a sample proportion and get results like 'between 55% and 60% agree with the statement,' this interval means we're 95% confident that the true population's agreement rate falls within this range. Using a 95% confidence level translates to a Z-score of 1.96, which is used in the formula for calculating the MOE to determine the interval limits accurately.