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Chapter 6: Problem 17

Suppose \(X \sim N(9,5) .\) What value of \(x\) has a \(z\) -score of \(-0.5 ?\)

Short Answer

Expert verified
The value of \(x\) with a \(z\)-score of \(-0.5\) is 6.5.

Step by step solution

01

Understand the Problem

We are given a normal distribution where the mean \(\mu\) is 9 and the standard deviation \(\sigma\) is 5. We need to find the value \(x\) that corresponds to a \(z\)-score of \(-0.5\).
02

Recall the Z-Score Formula

The \(z\)-score formula is \(z = \frac{x - \mu}{\sigma}\). This formula helps us find the difference between a data point \(x\) and the mean \(\mu\), scaled by the standard deviation \(\sigma\).
03

Plug in the Known Values

We know \(z = -0.5\), \(\mu = 9\), and \(\sigma = 5\). Substituting these into the formula gives us: \(-0.5 = \frac{x - 9}{5}\).
04

Solve for \(x\)

To solve for \(x\), first multiply both sides of the equation by 5: \(-0.5 \times 5 = x - 9\). This simplifies to: \(-2.5 = x - 9\).
05

Isolate \(x\)

Add 9 to both sides to isolate \(x\): \(-2.5 + 9 = x\). This results in \(x = 6.5\).

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Key Concepts

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

Z-score calculation
The Z-score is a statistical measure that tells us the position of a data point relative to the mean of a data set. It is expressed in terms of standard deviations.
Using the Z-score, we can determine how far and in what direction a data point deviates from the mean. To calculate it, we apply the formula:\[z = \frac{x - \mu}{\sigma},\] where:
  • \(z\) is the Z-score,
  • \(x\) is the data point of interest,
  • \(\mu\) is the mean of the distribution,
  • \(\sigma\) is the standard deviation.
A Z-score of 0 indicates that the data point is exactly at the mean, a positive Z-score means it's above the mean, and a negative Z-score means it's below the mean. Calculating the Z-score helps in standardizing different data sets to compare them easily.
Mean and standard deviation
In statistics, the mean and standard deviation are key components of data analysis.The mean, often denoted as \(\mu\), is the average of all data points in a data set. To find it, sum up all the data points and divide by the number of points. It serves as a central value within the data set.
The standard deviation, denoted by \(\sigma\), measures the amount of variation or dispersion of a set of values. It tells us how tightly the data points are clustered around the mean. To calculate the standard deviation:
  • First, find the variance by averaging the squared differences of each data point from the mean.
  • Then, take the square root of the variance to get the standard deviation.
A small standard deviation indicates that data points tend to be close to the mean, while a large standard deviation suggests a wide range of values.
Probability distributions
Probability distributions describe how the values of a random variable are distributed. They provide a map of the likelihood of each outcome. In the context of this exercise, the normal distribution is used, a kind of probability distribution thats symmetrical and bell-shaped.
The normal distribution is defined by its mean and standard deviation. Most data points are concentrated around the mean, tapering off symmetrically towards both ends. This gives the classic 'bell curve'.
  • The area under the curve corresponds to the probability of data falling within a certain range.
  • For a normal distribution, about 68% of data falls within one standard deviation of the mean, 95% within two, and 99.7% within three.
Understanding probability distributions makes it easier to anticipate the behavior of data sets and helps with statistical analysis and hypothesis testing.

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Most popular questions from this chapter

Facebook provides a variety of statistics on its Web site that detail the growth and popularity of the site. On average, 28 percent of 18 to 34 year olds check their Facebook profiles before getting out of bed in the morning. Suppose this percentage follows a normal distribution with a standard deviation of five percent. a. Find the probability that the percent of 18 to 34-year-olds who check Facebook before getting out of bed in the morning is at least 30. b. Find the 95th percentile, and express it in a sentence.

A NUMMI assembly line, which has been operating since 1984, has built an average of 6,000 cars and trucks a week. Generally, 10% of the cars were defective coming off the assembly line. Suppose we draw a random sample of n = 100 cars. Let X represent the number of defective cars in the sample. What can we say about X in regard to the 68-95-99.7 empirical rule (one standard deviation, two standard deviations and three standard deviations from the mean are being referred to)? Assume a normal distribution for the defective cars in the sample.

About what percent of the x values from a normal distribution lie within two standard deviations (left and right) of the mean of that distribution?

Use the following information to answer the next two exercises: The patient recovery time from a particular surgical procedure is normally distributed with a mean of 5.3 days and a standard deviation of 2.1 days. The heights of the 430 National Basketball Association players were listed on team rosters at the start of the 2005鈥�2006 season. The heights of basketball players have an approximate normal distribution with mean, ? = 79 inches and a standard deviation, ? = 3.89 inches. For each of the following heights, calculate the z-score and interpret it using complete sentences. a. 77 inches b. 85 inches c. If an NBA player reported his height had a z-score of 3.5, would you believe him? Explain your answer.

Use the following information to answer the next three exercises: The length of time it takes to find a parking space at 9 A.M. follows a normal distribution with a mean of five minutes and a standard deviation of two minutes. IQ is normally distributed with a mean of 100 and a standard deviation of 15. Suppose one individual is randomly chosen. Let X = IQ of an individual. a. X ~ _____(_____,_____) b. Find the probability that the person has an IQ greater than 120. Include a sketch of the graph, and write a probability statement. c. MENSA is an organization whose members have the top 2% of all IQs. Find the minimum IQ needed to qualify for the MENSA organization. Sketch the graph, and write the probability statement. d. The middle 50% of IQs fall between what two values? Sketch the graph and write the probability statement.
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