Question

In a survey of 1,000 randomly selected adults in the United States, participants were asked what their most favorite and least favorite subjects were when they were in school (Associated Press, August 17,2005\()\). In what might seem like a contradiction, math was chosen more often than any other subject in both categories. Math was chosen by 230 of the 1,000 as their most favorite subject and chosen by 370 of the 1,000 as their least favorite subject. a. Construct and interpret a \(95 \%\) confidence interval for the proportion of U.S. adults for whom math was their most favorite subject. b. Construct and interpret a \(95 \%\) confidence interval for the proportion of U.S. adults for whom math was their least favorite subject.

Short answer

The \(95 \% \) confidence interval for the proportion of U.S. adults for whom math was their most favorite subject and least favorite subject are obtained using the respective sample proportions and the provided confidence level. The interpretation is that we are \(95 \% \) confident that the true proportion lies within the calculated intervals.

Step-by-step solution

Calculate sample proportions

We first calculate the sample proportion (\(p\)) for both categories. This is done by dividing the number of successes (people who chose math as their favorite or least favorite subject) by the total sample size (1,000 in this case). So, for the favorite category, \(p_{Favorite} = \frac{230}{1000} = 0.23\). Similarly, for least favorite category, \(p_{Least} = \frac{370}{1000} = 0.37\).

Decide confidence level

The confidence level has already been decided in the question itself and it is \(95 \% \). This confidence level relates to \(1.96\) standard deviations in a normal distribution (Z-score for \(95 \% \) confidence level).

Calculate Confidence Interval

Use the formula for confidence interval: \(CI = p \pm Z*\sqrt{\frac{p(1-p)}{n}}\), where \( CI \) is confidence interval, \( p \) is sample proportion, \( Z \) is the Z-score, and \( n \) is the total sample size. Plugging in the values, we get the confidence interval in the 'most favorite' category as \( CI_{Favorite} = 0.23 \pm 1.96*\sqrt{\frac{0.23*(1-0.23)}{1000}} \), and for the 'least favorite' category, \(CI_{Least} = 0.37 \pm 1.96*\sqrt{\frac{0.37*(1-0.37)}{1000}} \).

Interpret the intervals

The final step is interpreting the confidence intervals. It means that we are \(95 \% \) confident that the true proportion of U.S. adults who chose math as their favorite subject lies within the calculated interval for 'most favorite', and similarly for 'least favorite'.

Key concepts

Sample Proportions

Understanding sample proportions is fundamental when dealing with any form of survey analysis. The sample proportion represents the fraction of the survey sample that exhibits a particular trait or characteristic. In the context of our exercise, where 1,000 U.S. adults were surveyed about their favorite and least favorite subjects in school, sample proportions come into play when calculating the percentages of those who preferred or disliked math.

For instance, if 230 out of 1,000 participants chose math as their most favorite subject, the sample proportion for this category is calculated by dividing 230 (the number of successes) by 1,000 (the total number of survey respondents), yielding a sample proportion of 0.23. This figure implies that 23% of the respondents in the sample favor math the most, which serves as an estimate for the larger population of U.S. adults.

Understanding these proportions is crucial, as they are the foundation upon which further analysis, such as confidence intervals, is built.

Z-score

The Z-score, often referred to as a standard score, measures how many standard deviations an element is from the mean. In surveys and statistical analysis, the Z-score is used to relate a sample statistic to the standard deviation of a sampling distribution. The Z-score thus plays a vital role in calculating confidence intervals for proportions.

In the given survey data, a Z-score corresponds to a confidence level. For a 95% confidence level, the Z-score is 1.96, meaning that the true population proportion is expected to lie within 1.96 standard deviations from the sample proportion 95% of the time. This Z-score is a constant figure taken from standard normal distribution tables, which map confidence levels to their respective Z-scores.

By using the Z-score, one can calculate a range (confidence interval) where we can expect the actual proportion to fall, given our sample data.

Survey Data Analysis

Survey data analysis involves examining data collected from a sample of individuals to make inferences about a larger population. In our example, the survey data consists of answers from 1,000 randomly chosen U.S. adults regarding their school subject preferences. This analysis typically includes calculating key statistics like sample proportions.

The analysis also often incorporates confidence intervals, which provide an estimated range of values likely to contain the population parameter (in this case, the proportion of U.S. adults preferring or disliking math). The width of these intervals can reveal the precision of the survey data: narrow intervals denote precise data, while wider intervals suggest more variability and less certainty.

In the example, constructing confidence intervals on the proportions helps ascertain the reliability of the survey findings and communicates the estimated proportions of the entire U.S. adult population that the survey represents.

Statistical Inference

Statistical inference involves drawing conclusions about a population based on data from a sample. It's a cornerstone of effective survey data analysis and involves estimating population parameters, testing hypotheses, and making predictions.

Regarding the school subject preference survey, we are using statistical inference to generalize our sample findings to the larger U.S. adult population. Confidence intervals are a tool of statistical inference that enable us to state, with a specific level of certainty (usually 95%), that the true proportion of the population with a particular preference is contained within a certain range.

The interpretation of these intervals is nuanced; a 95% confidence interval does not suggest that we are 95% sure that this exact interval contains the true proportion, but rather that if we were to repeat the study many times, 95% of the constructed intervals would contain the true proportion. Statistical inference hence imbues survey results with broader applicability and meaning, extending beyond the individuals immediately surveyed.