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Salaried workers at a large corporation receive 2 weeks' paid vacation per year. Sixteen randomly selected workers from this corporation were asked whether or not they would be willing to take a \(3 \%\) reduction in their annual salaries in return for 2 additional weeks of paid vacation. The following are the responses of these workers. \(\begin{array}{llllllll}\text { No } & \text { Yes } & \text { No } & \text { No } & \text { Yes } & \text { No } & \text { No } & \text { Yes } \\ \text { Yes } & \text { No } & \text { No } & \text { No } & \text { Yes } & \text { No } & \text { No } & \text { No }\end{array}\) Construct a \(97 \%\) confidence interval for the percentage of all salaried workers at this corporation who would accept a \(3 \%\) pay cut in return for 2 additional weeks of paid vacation.

Step by step solution

01

Determine Proportions

Identify the total number of respondents and the number who answered 'yes'. The total number of respondents indicated in the exercise is 16. The responses 'yes' are 4 out of 16. Hence the proportion of 'yes' responses (\(p\)) is \(4/16 = 0.25\)

02

Apply Confidence Interval Formula

Apply the formula for a confidence interval for a proportion, which is \(p \pm Z*\sqrt{p(1-p)/n}\) where; • \(p = 0.25\), the proportion of 'yes' responses,• \(Z\) is the Z-score corresponding to the desired confidence level (in this case, 97%), which from statistical tables is approximately \(2.17\), • \(n = 16\), the total number of observations. Substitute the values into the formula to get: \(0.25 \pm (2.17*\sqrt{0.25*0.75/16})\)

03

Calculate the Confidence Interval

Carry out the calculation in the formula of Step 2: The result of \(2.17*\sqrt{0.25*0.75/16}\) is approximately \(0.203\). Therefore the range for the confidence interval is \(0.25 \pm 0.203\), which gives a lower boundary of \(0.047\) and an upper boundary of \(0.453\). Thus, one can be 97% confident that the true proportion of salaried workers who would accept the deal lies within this interval.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Proportion

The concept of proportion in statistics is crucial when dealing with categorical data. A proportion represents a part of a total, expressed as a fraction or percentage. In the exercise, the proportion refers to the ratio of salaried workers who agreed to take a pay cut in exchange for more vacation time. For our exercise, we counted 4 'Yes' responses out of a total of 16 workers surveyed, giving us a proportion, denoted as \( p \), of \( \frac{4}{16} = 0.25 \) or 25%. This means 25% of the sampled employees agreed to the salary reduction for an additional vacation period. Understanding the proportion helps us quantify the opinion or behavior of a subset of data within the larger data set or population.

Z-score

In statistical analysis, the Z-score allows us to determine the number of standard deviations a data point is from the mean. For confidence intervals, Z-scores illustrate how confident we are about our population parameter estimates. To determine the Z-score relevant for a confidence interval, you need to select the confidence level first. For a 97% confidence level, statistical tables are used to find a Z-score, which in our case is approximately \(2.17\). This value indicates that we are 97% confident the actual population parameter falls within the calculated range. Z-scores are fundamental in hypothesis testing and calculating confidence intervals because they adjust based on the desired level of confidence, ensuring the results are statistically valid.

Statistical Tables

Statistical tables, particularly Z-tables, are essential references used to find the corresponding Z-score for different confidence levels. These tables provide the probability (or percentage) that a statistic is below a certain Z-value in a standard normal distribution.In our exercise, to construct a 97% confidence interval, we consulted a Z-table to find that the Z-score is approximately \(2.17\). This tells us that 97% of our data falls within this range when normalized, ensuring our interval estimation accounts for variability and provides a trustworthy approximation.Using statistical tables effectively allows statisticians to bypass complex calculations, relying instead on established values for common distributions like the standard normal distribution.

Sample Size

Sample size, denoted as \( n \), is the number of observations in a sample. It significantly influences the accuracy and reliability of statistical inferences. In our scenario, the sample size is 16, representing the number of surveyed workers.A larger sample size leads to more stable outcomes and narrower confidence intervals because it captures more variation and provides a more accurate estimate of the population parameter. Conversely, a small sample size can lead to wider confidence intervals and less precise estimates, highlighting potential sampling error.Understanding how sample size impacts statistical tests is important in designing studies and interpreting results, ensuring conclusions are based on representative data that accurately reflect the population.