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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. $$\begin{aligned}&\text { Descriptive Statistics }\\\&\begin{array}{rrrr}\mathrm{N} & \text { Event } & \text { Sample p } & 95 \% \mathrm{Cl} \text { for } \mathrm{p} \\\\\hline 1008 & 675 & 0.669643 & (0.639648,0.698643)\end{array}\end{aligned}$$ 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, it is noticed that confidence interval for the population proportion is (0.639648,0.698643). These values are in decimal form. To convert them into a percentage form, each value needs to be multiplied by 100. So, the range of confidence interval becomes (63.96 %, 69.86 %). Therefore, with the 95% confidence level, it can be said that the percentage of the population believing that the colleges and Universities with big sports programs place too much emphasis on athletics over academics is between 63.96 % and 69.86 %.

02

Confirm the Blogger's Claim

According to the claim of a sports blogger, the majority of Americans believe that the colleges and Universities with big sports programs place too much emphasis on athletics over academics. In data analysis, the majority is usually considered as more than 50%. From the confidence interval (63.96 %, 69.86 %), it can be observed that both values are above 50%, which represents the majority. So, the blogger's claim is supported by the given confidence interval.

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Key Concepts

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

Descriptive Statistics

Understanding the power of descriptive statistics is essential in interpreting data and drawing meaningful conclusions. Descriptive statistics provide us with a summary about the sample data, giving us a quick insight about the distribution, tendencies, and spread. In the Monmouth University poll, the key piece of descriptive statistics is the sample proportion, denoted as 'Sample p', which tells us the proportion of individuals in the sample who believe that colleges with big sports programs prioritize athletics over academics. This is represented by the value 0.669643, or 66.96% when expressed as a percentage.

Descriptive statistics also include measures of central tendency, like mean and median, as well as measures of variability such as range, variance, and standard deviation. However, for this problem, the focus is on the proportion, which is a central aspect of the poll's findings. It's important for students to not just calculate these statistics, but also to comprehend what they imply about the population in question.

Population Proportion

The population proportion is a key concept in statistics and represents the ratio of individuals in the entire population that have a specified characteristic, in this case, the belief about sports programs' emphasis on athletics over academics. The school exercise required translating a sample proportion into a broader context to estimate the population proportion. This proportion gives us an insight into the larger group's behavior or opinion based on our sample.

When we speak of a '95% confidence interval', it entails that if we were to take 100 different samples and calculate 100 confidence intervals, we expect that 95 of those 100 intervals will contain the true population proportion. Therefore, we can express in the given problem that we are 95% confident that the true proportion of all Americans sharing the belief falls between 63.96% and 69.86%. Not only does this give us a statistical estimate, but it also communicates the reliability and potential accuracy of our result.

Central Limit Theorem (CLT)

The Central Limit Theorem (CLT) is a fundamental principle in statistics that aids in understanding the distributions of sample statistics. For large enough sample sizes, the CLT tells us that the sampling distribution of the sample statistic (like the sample mean or sample proportion) will be approximately normally distributed, even if the population distribution is not normal. This theorem is incredibly powerful because it allows us to make inferences about population parameters using sample data.

Now, let's consider the Monmouth University poll in relation to the CLT. The assumption that 'the conditions for using the CLT are met' implies that the sample size is large enough for the sampling distribution of the sample proportion to be considered normal. This condition legitimizes the construction of a confidence interval for the population proportion. The poll's sample size of 1,008 individuals is indeed large enough to invoke the CLT, resulting in a useful approximation for making inferences about the broader population's views on college sports programs.