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Chapter 7: Problem 7

Use a normal probability plot to assess whether the sample data could have come from a population that is normally distributed. O-Ring Thickness A random sample of O-rings was obtained, and the wall thickness (in inches) of each was recorded. $$\begin{array}{lllll} \hline 0.276 & 0.274 & 0.275 & 0.274 & 0.277 \\ \hline 0.273 & 0.276 & 0.276 & 0.279 & 0.274 \\ \hline 0.273 & 0.277 & 0.275 & 0.277 & 0.277 \\ \hline 0.276 & 0.277 & 0.278 & 0.275 & 0.276 \\ \hline \end{array}$$

Short Answer

Expert verified
If points on the plot closely align along a straight line, the data likely come from a normally distributed population.

Step by step solution

01

Gather and Sort the Data

List the data points in ascending order: 0.273, 0.273, 0.274, 0.274, 0.274, 0.274, 0.275, 0.275, 0.275, 0.276, 0.276, 0.276, 0.276, 0.276, 0.277, 0.277, 0.277, 0.277, 0.277, 0.278, 0.279.
02

Compute the Expected Z-Scores

For each data point, compute the expected z-scores assuming a normal distribution. These can be found using a z-score table or statistical software.
03

Create the Normal Probability Plot

On the x-axis, place the expected z-scores obtained in Step 2. On the y-axis, place the sorted observed data. Plot each pair of observed data and expected z-score.
04

Analyze the Plot

Observe the plot to assess normality. If the data points fall on or near a straight line, the sample likely comes from a normally distributed population. Deviations from a straight line indicate non-normality.

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Key Concepts

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

Assessing Normality
When analyzing sample data, it鈥檚 crucial to determine if the data follows a normal distribution. Assessing normality can help in choosing the appropriate statistical tests and in drawing accurate conclusions.

A normal probability plot is a powerful tool used to check if a dataset approximates a normal distribution. The idea is simple: if the data is normally distributed, the points on the plot will follow a straight line.

In this exercise, we have O-ring thickness data. By plotting this data on a normal probability plot, we assess its normality. Deviations from the straight line indicate the data is not normally distributed.
Z-Scores
Z-scores play a vital role in determining the normality of data. They measure how many standard deviations a data point is from the mean.

To create a normal probability plot, we calculate the expected z-scores for the sorted data. These z-scores represent the position that each data point would likely achieve if the data were perfectly normally distributed.

Using statistical software or a z-score table simplifies this process. Each data point then pairs with its corresponding expected z-score, ready for plotting.
Normal Distribution
A normal distribution, also known as a Gaussian distribution, is symmetric and bell-shaped. Most of the data points are close to the mean, with fewer points as you move away.

Normal distributions are foundational in statistics because they describe many natural phenomena, and many statistical tests assume normality.

Understanding the normal distribution helps in interpreting the results of your data analysis. When the data is normally distributed, you can use z-scores and probability plots to identify outliers and make predictions about data behavior.
Sample Data Analysis
Analyzing sample data involves several steps, each critical to ensuring accurate results.

For our O-ring thickness data:
  • We first gather and sort the data points in ascending order.
  • Next, we calculate the expected z-scores for each data point, assuming a normal distribution.
  • We then create a normal probability plot by pairing each observed data point with its expected z-score and plotting these pairs.
  • Finally, we analyze the plot. If the points form a straight line, the data is likely normally distributed. Deviations suggest non-normality.


By following these steps, you can effectively assess the normality of any dataset, leading to more accurate and reliable statistical conclusions.

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Most popular questions from this chapter

Use the results in the table to (a) draw a normal probability plot, (b) determine the linear correlation between the observed values and expected z-scores, (c) determine the critical value in Table VI to assess the normality of the data $$ \begin{array}{cccc} \text { Index, } i & \text { Observed Value } & f_{i} & \text { Expected } z \text { -score } \\ \hline 1 & 1 & 0.09 & -1.34 \\ \hline 2 & 3 & 0.22 & -0.77 \\ \hline 3 & 6 & 0.36 & -0.36 \\ \hline 4 & 8 & 0.50 & 0 \\ \hline 5 & 10 & 0.64 & 0.36 \\ \hline 6 & 13 & 0.78 & 0.77 \\ \hline 7 & 35 & 0.91 & 1.34 \end{array} $$

Find the indicated areas. For each problem, be sure to draw a standard normal curve and shade the area that is to be found. Determine the area under the standard normal curve that lies between (a) \(z=-2.55\) and \(z=2.55\) (b) \(z=-1.67\) and \(z=0\) (c) \(z=-3.03\) and \(z=1.98\)

The number of chocolate chips in an 18 -ounce bag of Chips Ahoy! chocolate chip cookies is approximately normally distributed, with a mean of 1262 chips and a standard deviation of 118 chips, according to a study by cadets of the U.S. Air Force Academy. Source: Brad Warner and Jim Rutledge, Chance \(12(1): 10-14,1999\) (a) Determine the 30 th percentile for the number of chocolate chips in an 18 -ounce bag of Chips Ahoy! cookies. (b) Determine the number of chocolate chips in a bag of Chips Ahoy! that make up the middle \(99 \%\) of bags. (c) What is the interquartile range of the number of chips in Chips Ahoy! cookies?

The lengths of human pregnancies are approximately normally distributed, with mean \(\mu=266\) days and standard deviation \(\sigma=16\) days. (a) What proportion of pregnancies lasts more than 270 days? (b) What proportion of pregnancies lasts less than 250 days? (c) What proportion of pregnancies lasts between 240 and 280 days? (d) What is the probability that a randomly selected pregnancy lasts more than 280 days? (e) What is the probability that a randomly selected pregnancy lasts no more than 245 days? (f) A "very preterm" baby is one whose gestation period is less than 224 days. Are very preterm babies unusual?

Assume that the random variable \(X\) is normally distributed, with mean \(\mu=50\) and standard deviation \(\sigma=7 .\) Compute the following probabilities. Be sure to draw a normal curve with the area corresponding to the probability shaded. \(P(38
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