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Chapter 7: Problem 13

Use a normal probability plot to assess whether the sample data could have come from a population that is normally distributed. A random sample of 20 undergraduate students receiving student loans was obtained, and the amount of their loans for the \(2004-2005\) school year was recorded. $$\begin{array}{ccccc}2,500 & 1,000 & 2,000 & 14,000 & 1,800 \\\\\hline 3,800 & 10,100 & 2,200 & 900 & 1,600 \\ \hline 500 & 2,200 & 6,200 & 9,100 & 2,800 \\\\\hline 2,500 & 1,400 & 13,200 & 750 & 12,000\end{array}$$

Short Answer

Expert verified
Perform a normal probability plot; if points form a straight line, data is normally distributed.

Step by step solution

01

- Organize the Data

List the amounts of student loans in ascending order: 500, 750, 900, 1000, 1400, 1600, 1800, 2000, 2200, 2200, 2500, 2500, 2800, 3800, 6200, 9100, 10100, 12000, 13200, 14000.
02

- Determine Percentiles

Find the percentiles for the corresponding ranks \textbf{Percentile} = \(\frac{rank-0.5}{N}\). Here, the rank ranges from 1 to 20, and N is 20. Calculate each percentile.
03

- Compute Z-Scores

Using the standard normal distribution table, find the Z-scores for the calculated percentiles.
04

- Plot the Data

Create a normal probability plot by plotting the Z-scores (on the y-axis) against the ordered loan amounts (on the x-axis).
05

- Analyze the Plot

Examine the plot. If the points roughly form a straight line, the sample data can be said to come from a normally distributed population.

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Key Concepts

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

Data Distribution
Understanding data distribution is critical for statistical analysis. It tells us how data points are spread across different values. A data distribution can show patterns, trends, and probabilities. In this context, organizing the data, such as the amounts of student loans, helps in visualizing the spread and understanding key characteristics.
  • Symmetric: Data values are evenly spread on both sides of the mean.
  • Skewed: Data values lean more towards one side, right or left.
  • Uniform: Every data value has the same frequency of occurrence.
  • Normal: Data points are symmetrically distributed around the mean, forming a bell-shaped curve.
When using a normal probability plot, visually inspecting the organized data helps determine if it is normally distributed.
Z-Scores
Z-scores, also known as standard scores, help compare different data points in terms of how many standard deviations they are from the mean. For any data value, the Z-score formula is:
\[ Z = \frac{X - \mu}{\sigma} \]
Where:
  • \text{X} is the data point.
  • \text{\mu} is the mean of the data set.
  • \text{\sigma} is the standard deviation of the data set.
In the exercise, you use the percentile to find Z-scores using the standard normal distribution table. These Z-scores help in plotting the data against the ideal normal distribution. If the data aligns well along a straight line on this plot, it suggests the data is normally distributed.
Percentiles
Percentiles rank the position of a certain data point compared to the entire data set. They tell us the percentage of data points lying below a particular value. To find percentiles for ranked data:
\[ \text{Percentile} = \frac{\text{rank} - 0.5}{N} \]
Where:
  • \text{rank} represents the position of the data point when arranged in ascending order, and
  • \text{N} represents the total number of data points.
For instance, in a sample of 20 student loan amounts, the percentile helps you understand how each loan amount compares to the rest. This step is essential for normal probability plotting, and calculating these percentiles allows for finding the corresponding Z-scores.
Statistical Analysis
Statistical analysis involves collecting, organizing, interpreting, and presenting data in a meaningful way. It helps in making informed decisions based on data. The normal probability plot is a statistical tool used to assess if sample data comes from a normally distributed population. The steps include:
  • Organizing the data.
  • Finding percentiles.
  • Calculating Z-scores.
  • Creating a normal probability plot.
  • Analyzing the plot.
By plotting the Z-scores against the ordered data, you visually determine normality. A straight-line formation indicates normal distribution, while deviations suggest otherwise. This analysis provides valuable insights for further research and understanding patterns within the collected data.

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Most popular questions from this chapter

Suppose the life of refrigerators is normally distributed with mean \(\mu=14\) years and standard deviation \(\sigma=2.5\) years. (Source: Based on information obtained from Consumer Reports) (a) Draw a normal curve with the parameters labeled. (b) Shade the region that represents the proportion of refrigerators that are kept for more than 17 years. (c) Suppose the area under the normal curve to the right of \(X=17\) is \(0.1151 .\) Provide two interpretations of this result.

Earthquakes The magnitude of earthquakes since 1900 that measure 0.1 or higher on the Richter scale in California is approximately normally distributed, with \(\mu=6.2\) and \(\sigma=0.5,\) according to data obtained from the U.S. Geological Survey. (a) What is the probability that a randomly selected earthquake in California has a magnitude of 6.0 or higher? (b) What is the probability that a randomly selected earthquake in California has a magnitude less than \(6.4 ?\) (c) What is the probability that a randomly selected earthquake in California has a magnitude between 5.8 and \(7.1 ?\) (d) The great San Francisco Earthquake of 1906 had a magnitude of \(8.25 .\) Is an earthquake of this magnitude unusual in California? (e) What is the percentile rank of a California earthquake that measures 6.8 on the Richter scale? (f) What is the percentile rank of a California earthquake that measures 5.1 on the Richter scale?

Under what circumstances can the normal distribution be used to approximate binomial probabilities?

Compute \(P(x)\) using the binomial probability formula. Then determine whether the normal distribution can be used as an approximation for the binomial distribution. If so, approximate \(P(x)\) and compare the result to the exact probability. $$n=75, p=0.75, X=60$$

In a 2003 study, the Accreditation Council for Graduate Medical Education found that medical residents' mean number of hours worked in a week is \(81.7 .\) Suppose the number of hours worked per week by medical residents is approximately normally distributed with a standard deviation of 6.9 hours. (Source: www.medrecinst.com) (a) Determine the 75 th percentile for the number of hours worked in a week by medical residents. (b) Determine the number of hours worked in a week that makes up the middle \(80 \%\) of medical residents.
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