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A recent Monmouth University poll found that 675 out of 1008 randomly selected people in the United States felt that college and universities with big sports programs placed too much emphasis on athletics over academics. Assuming the conditions for using the CLT were met, use the Minitab output provided to answer these questions. $$ \begin{aligned} &\text { Descriptive Statistics }\\\ &\begin{array}{rrrr} \mathrm{N} & \text { Event } & \text { Sample } \mathrm{p} & 95 \% \mathrm{Cl} \text { for } \mathrm{p} \\ \hline 1008 & 675 & 0.669643 & (0.639648,0.698643) \end{array} \end{aligned} $$ a. Complete this sentence: I am \(95 \%\) confident that the population proportion believing that colleges and universities with big sports programs place too much emphasis on athletics over academics is between _____ and _____. Report each number as a percentage rounded to one decimal place. b. Suppose a sports blogger wrote an article claiming that the majority of Americans believe that colleges and university with big sports programs place too much emphasis on athletics over academics. Does this confidence interval support the blogger's claim? Explain your reasoning.

Step by step solution

01

Interpret the Confidence Interval

From the Minitab output provided, we can see that the 95% confidence interval for the population proportion (\(p\)) is given as (0.639648, 0.698643). The confidence interval suggests that, if we were to repeat this survey many times, we would expect the population proportion to fall within this interval in 95% of the samples drawn.

02

Answer Part A

The question asks for the interpretation of the confidence interval in percentage terms. To convert the interval to percentages, we multiply the lower and upper bounds by 100: \([0.639648*100, 0.698643*100] = [63.96, 69.86]\). We are therefore 95% confident that the population proportion believing that colleges and universities with big sports programs place too much emphasis on athletics over academics is between 63.96% and 69.86%, rounded to one decimal place, is between 64.0% and 69.9%.

03

Evaluate a Claim (Part B)

The sports blogger's claim is that the majority of Americans believe that colleges and university with big sports programs place too much emphasis on athletics over academics. Given that 'majority' typically means more than half, or 50%, we can say that this claim is supported by the confidence interval as both the lower and upper bounds (64.0% and 69.9%) are greater than 50%.

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Key Concepts

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

Central Limit Theorem

The Central Limit Theorem (CLT) is a fundamental principle in statistics that assists in making inferences about a population based on sample data. What makes CLT so powerful is its simplicity: no matter the shape of the population distribution, the distribution of the sample means will approximate a normal distribution as the sample size becomes large enough.
This theorem allows us to apply the same confidence intervals and statistical tests to sample means even when we do not know the underlying population distribution. For this to hold best, the sample size should be sufficiently large, typically n > 30 is considered adequate in practice.

• The CLT allows researchers to draw conclusions about population parameters.
• It establishes the groundwork for constructing confidence intervals as used in the exercise.
• It helps validate the assumption underlying the computation of confidence intervals, suggesting that the calculated intervals are reliable.

Given the context of the Monmouth University poll, the validity of the 95% confidence interval relies heavily on the CLT, ensuring that our sample of 1008 people is large enough to give us a dependable approximation of the population proportion.

Population Proportion

Population proportion, denoted as \( p \), is a statistical metric that defines the fraction of a total population that exhibits a particular attribute. In surveys and poll analysis, such as the one conducted by Monmouth University, sample proportions are used to estimate population proportions. This involves using a sample (like the 1008 individuals surveyed) to infer about the larger group in the entire country.
Understanding population proportion is critical because:

• It provides an estimate of how prevalent a particular trait or opinion is across a population.
• It helps in decision-making processes, including policy setting and strategic planning.
• Accurate estimation of population proportion helps confirm if findings from the sample are reflective of the broader population.

In the exercise, the sample proportion was used to determine the range (confidence interval) within which the true population proportion is expected to fall. This tells us how likely it is that our observed data accurately represents the real-world scenario.

Statistical Inference

Statistical inference is all about drawing conclusions about a population based on data from a sample. It involves methods like confidence intervals and hypothesis testing, which allow us to make reasonable guesses or decisions without examining an entire population.
The exercise you're dealing with uses confidence intervals as a form of inference, helping us make informed statements about the population:

• Confidence intervals present a range of values within which we can be fairly certain a population parameter lies.
• They consider variability and sample size to provide context on how reliable the results are.
• In scenarios like this, an interval estimate is preferred to a single point estimate because it conveys the margin of error, giving more insight into our estimate's accuracy.

More broadly, statistical inference relies on the evaluation of the data against our hypotheses. For example, evaluating the blogger's claim involves determining if the confidence interval aligns with the notion that a majority (over 50%) indeed hold a particular belief.