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College costs rise" (October 29,2008 ), an article on the CNN Money website, gave the latest figures from the College Board on annual tuition, fees, and room and board. The average total figures are \(\$ 34,132\) for private colleges and \(\$ 14,333\) for public colleges. In an effort to compare those same costs in New York State, a sample of 32 junior students was randomly selected statewide from the private colleges and 32 more from the public colleges. The private college sample resulted in a mean of \(\$ 34,020,\) and the public college sample mean was \(\$ 14,045\) a. Assuming the annual college fees for private colleges have a mounded distribution and the standard deviation is \(\$ 2200,\) find the \(95 \%\) confidence interval for the mean college costs. b. Assuming the annual college fees for public colleges have a mounded distribution and the standard deviation is \(\$ 1500,\) find the \(95 \%\) confidence interval for the mean college costs. c. How do the New York State college costs compare to the College Board's values? Explain. d. Compare the confidence intervals found in parts a and b and describe the effect the two different sample means had on the resulting answers. e. Compare the confidence intervals found in parts a and b and describe the effect the two different sample standard deviations had on the resulting answers.

Step by step solution

01

Calculate the Confidence Interval for Private Colleges

First, let's calculate the 95% confidence interval for private colleges using the formula: \(CI = \bar{x} \pm z \cdot \frac{s}{\sqrt{n}}\), where \(\bar{x}=34020\), \(s=2200\), \(n=32\), and the z-score for a 95% CI is 1.96. Substituting these values into the formula, we get: \(CI = 34020 \pm 1.96 \cdot \frac{2200}{\sqrt{32}}\).

02

Calculate the Confidence Interval for Public Colleges

Next, let's calculate the 95% confidence interval for public colleges similarly. The sample mean is \(\bar{x}=14045\), \(s=1500\), \(n=32\), and we use the same 1.96 for the z-score. Plugging in these values, \(CI = 14045 \pm 1.96 \cdot \frac{1500}{\sqrt{32}}\).

03

Compare New York State College Costs with College Board's Values

Once the confidence intervals are found, we can compare these with the given College Board's average total figures: \(34132\) for private colleges and \(14333\) for public colleges.

04

Effect of Different Sample Means on the Confidence Intervals

The effect of different sample means on the confidence interval can be analyzed by comparing the intervals found in steps 1 and 2. Different sample means shift the centre of the confidence interval.

05

Effect of Different Standard Deviations on the Confidence Intervals

The effect of different standard deviations can also be analyzed by comparing the confidence intervals found in steps 1 and 2. The standard deviation impacts the width of the confidence interval. A larger standard deviation results in a wider confidence interval.

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Key Concepts

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

College Costs

Understanding college costs is essential for students and families planning higher education. In this exercise scenario, we are exploring college costs through the average annual tuition, fees, and room and board for private and public colleges.
From the data, the average is \\(34,132\ for private colleges and \\)14,333\ for public colleges, according to the College Board.
To compare these figures with those specific to New York State, samples from 32 private college students and 32 public college students are used.

• The private college student sample reported an average of \\(34,020\, while the public college sample showed an average of \\)14,045\.
• These sample means are critical for constructing confidence intervals, which help us assess how close these sample estimates are to the true population means.

With the calculated confidence intervals, we can statistically infer how New York State college costs measure up to the broader College Board's data.
This illustration serves as a practical application of statistical methods in real-world financial planning in educational contexts.

Standard Deviation

Standard deviation is a measure of the spread or variability of a set of data points.
In the context of this exercise, it helps us understand the variability in college costs among students.
For private colleges, the standard deviation is \\(2200\, indicating how much the individual costs tend to vary around the mean cost of \\)34,020\.

• For public colleges, the standard deviation is \\(1500\.
• The smaller standard deviation for public colleges signifies that the costs are more concentrated around the mean of \\)14,045\, compared to private colleges.

In calculating the confidence intervals, the standard deviation plays a crucial role as it directly influences the interval's width.
A larger standard deviation results in a wider confidence interval, suggesting more uncertainty about the true mean.
Conversely, a smaller standard deviation produces a tighter confidence interval, indicating more confidence in the sample mean estimation.

Sample Means

Sample means represent the average value within a sample and are used as point estimates of the population mean.
In this exercise, the average costs for the private college sample and the public college sample are \\(34,020\ and \\)14,045\, respectively.

• These sample means are utilized to calculate the confidence intervals indicated in the exercise steps.
• The distance between the sample mean and the College Board's reported averages highlights the importance of sample means in representing data accurately.

When constructing confidence intervals, these sample means act as the central point around which the interval is built.
Thus, they shift the entire interval up or down, depending on how close or far the sample mean is from any central tendency such as the College Board's figures.
Understanding the position of sample means aids in determining their reliability in reflecting the actual cost figures.

Mounded Distribution

A mounded distribution refers to data that is symmetrically distributed around a central peak, resembling a bell-shaped curve.
In this exercise, the assumption that college costs have a mounded distribution is crucial for the application of statistical methods, such as z-scores and confidence intervals.

• This assumption is particularly vital when employing the normal distribution, which many statistical analyses, including confidence intervals, rely on.
• The shape of the mounded distribution suggests that most data points, like student college costs, cluster around the mean with fewer instances as you move away from the center.

Given the assurance of a mounded distribution, we can confidently use standard deviation and z-scores to assess the confidence intervals.
This assumption affirms that the confidence intervals constructed are valid reflections of where the true mean comfortingly lies in relation to its spread across different samples.