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Suppose that a campus bookstore manager wants to know the proportion of students at the college who purchase some or all of their textbooks online. Two different people independently selected random samples of students at the college and used their sample data to construct the following confidence intervals for the population proportion: Interval 1:(0.54,0.57) Interval 2:(0.46,0.62) a. Explain how it is possible that the two confidence intervals are not centered in the same place. b. Which of the two intervals conveys more precise information about the value of the population proportion? c. If both confidence intervals have a \(95 \%\) confidence level, which confidence interval was based on the smaller sample size? How can you tell? d. If both confidence intervals were based on the same sample size, which interval has the higher confidence level? How can you tell?

Step by step solution

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a. Confidence interval centering

Confidence intervals are not centered in the same place because they have been calculated using two separate random samples of students. These samples may have different proportions of students who purchase their textbooks online, which leads to the different centering of the confidence intervals. Random sampling can sometimes result in slightly different findings due to the randomness of the sampling process.

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b. Precise information about the population proportion

To determine which confidence interval conveys more precise information, we can look at their width. Narrower intervals are more precise than wider ones. Interval 1: 0.57 - 0.54 = 0.03 Interval 2: 0.62 - 0.46 = 0.16 Since Interval 1 is narrower than Interval 2, it provides more precise information about the value of the population proportion.

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c. Smaller sample size for 95% confidence level

If both confidence intervals have a 95% confidence level, the interval with the larger width was likely based on the smaller sample size. This is because the width of a confidence interval is determined by two factors: (1) the underlying variation in the population, and (2) the sample size. So, for the same confidence level, if the interval is wider, the sample size is probably smaller. Based on our previous calculations, Interval 2 is wider than Interval 1. Therefore, Interval 2 was likely based on the smaller sample size.

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d. Higher confidence level for the same sample size

If both confidence intervals were based on the same sample size, the interval with the larger width would have a higher confidence level. This is because the confidence level indicates how likely it is that the true population proportion is contained within the interval. A higher confidence level will require a wider interval to cover more possibilities for the true population proportion. Since Interval 2 is wider than Interval 1, Interval 2 has a higher confidence level if both intervals were based on the same sample size.

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Key Concepts

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

Population Proportion

The population proportion represents the fraction of the population that has a certain characteristic. For instance, when a campus bookstore manager is interested in knowing the rate at which college students purchase their textbooks online, this rate is the population proportion in question. Since it's not feasible to survey every single student, a sample is taken to estimate this proportion. However, since each sample may include different segments of the population, it could lead to variations in the estimated proportion. This is why different random samples can generate different intervals, as it happened in the provided textbook problem.

Sample Size

Sample size refers to the total number of observations or individuals included in a sample. It's a critical aspect of statistical analysis because the size of the sample closely affects the level of confidence one might have in the resulting estimates. A larger sample size generally leads to a narrower confidence interval, which implies a more precise estimate of the population parameter—a concept illustrated in the original problem where the narrower Interval 1 suggests it was based on a larger sample size than the wider Interval 2.

Random Sampling

Random sampling is a fundamental technique used in statistics to select a subset of individuals from a larger population in a way that every individual has an equal chance of being chosen. This process is essential to obtain a representative sample, which can be used to make estimations about the population as a whole. However, random sampling is also susceptible to sampling variation which can account for the different confidence intervals that the two independent samples provided in the bookstore example.

Statistical Precision

Statistical precision pertains to how closely the sample estimate reflects the true population value. In terms of confidence intervals, precision is indicated by the width of the interval. A more precise interval is narrower, suggesting that the sample's estimation of the population parameter is more accurate. This concept is demonstrated when comparing the two confidence intervals in the example, where Interval 1 is narrower (hence more precise) than the broader Interval 2.

Confidence Level

The confidence level indicates the probability that the true population parameter lies within the confidence interval. Often expressed as a percentage, such as 95%, it shows our degree of certainty in the interval estimate. The student bookstore problem shows two confidence intervals with a 95% confidence level. This means we can say with 95% certainty that the true population proportion of students purchasing textbooks online falls within these ranges. However, for the same sample size, a wider interval implies a higher confidence level, and a narrower interval represents a lower confidence level.