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Chapter 6: Problem 32

Refer to the normal distributions for women's height \((\mu=65, \sigma=3.5)\) and men's height \((\mu=70, \sigma=4.0) .\) A man's height of 75 inches and a woman's height of 70 inches are both 5 inches above their means. Which is relatively taller? Explain why.

Short Answer

Expert verified
The woman is relatively taller because her height's z-score (1.43) is greater than the man's (1.25).

Step by step solution

01

Understanding the Concept

To determine which individual is relatively taller, we need to use the concept of the z-score. The z-score tells us how many standard deviations an observation is from the mean.
02

Calculate the Man's Z-Score

The z-score for the man's height is calculated with the formula: \[ z = \frac{X - \mu}{\sigma} \] where \( X \) is the height, \( \mu \) is the mean, and \( \sigma \) is the standard deviation. For the man, \( X = 75, \mu = 70, \sigma = 4 \). Thus,\[ z_{man} = \frac{75 - 70}{4} = \frac{5}{4} = 1.25 \]
03

Calculate the Woman's Z-Score

Similarly, compute the z-score for the woman's height: \[ z_{woman} = \frac{70 - 65}{3.5} = \frac{5}{3.5} \approx 1.43 \]
04

Compare the Z-Scores

The z-score tells us how far the height is from the mean in terms of standard deviations. The z-score for the man is 1.25, and for the woman, it is approximately 1.43. Since 1.43 is greater than 1.25, the woman's height is relatively taller compared to the typical distribution of women's heights.

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Key Concepts

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

Understanding Normal Distribution
The concept of a normal distribution is central to statistics. It refers to a probability distribution that is symmetric around its mean, forming a bell-shaped curve. Most of the data points lie close to the mean, and the likelihood of extreme values decreases as you move away from it. In a normal distribution:
  • The mean (average) is the center of the distribution.
  • The standard deviation tells us about the spread of the data.
  • Approximately 68% of the data falls within one standard deviation from the mean.
  • About 95% falls within two standard deviations.
  • Nearly 99.7% falls within three standard deviations.
This distribution often models natural phenomena, like heights or test scores, making it a powerful tool to analyze and predict outcomes.
Deciphering Standard Deviation
Standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a set of values. It's key to understanding how much individual data points typically differ from the mean. A low standard deviation means that the values tend to be close to the mean. On the other hand, a high standard deviation indicates that the values are spread out over a wider range. In formulas like the z-score, which shows how many standard deviations away a certain value is from the mean, the standard deviation helps standardize different sets of data. By using the standard deviation, you can make direct comparisons between different distributions.
Exploring the Meaning of Mean
The mean is commonly understood as the average of a set of numbers and is calculated by adding all the numbers together and dividing by the total count. It provides a central value around which the data is distributed. In normal distribution, this central value splits the data symmetrically.
  • The mean gives a quick summary of data but doesn't indicate spread or variability.
  • As a measure of central tendency, it is complemented by other metrics like median and mode.
  • In the context of normal distribution, the mean, together with the standard deviation, helps paint the full picture of the data set.
Understanding the mean is crucial for distinguishing whether an individual observation is typical for its distribution or if it stands out.
Conducting Comparative Analysis
Comparative analysis is a method used to compare different data sets and draw meaningful conclusions. In the context of this exercise, we performed comparative analysis using z-scores. A z-score provides a way to compare values from different distributions by converting them into a "standardized" form. Here's how it works:
  • Calculate the z-score to understand an observation's relative position within its data set.
  • Observe which z-score is higher鈥攈igher z-scores indicate observations that are further from the mean relative to their standard deviations.
  • This allows for easy comparison across different distributions.
By converting raw data into z-scores, not only can we see which observation is more extreme, but we can also do so across different groups, like men's and women's heights in this case.

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Most popular questions from this chapter

The normal distribution for women's height in North America has \(\mu=65\) inches, \(\sigma=3.5\) inches. Most major airlines have height requirements for flight attendants (www.cabincrewjobs.com). Although exceptions are made, the minimum height requirement is 62 inches. What proportion of adult females in North America are not tall enough to be a flight attendant?

The state of Ohio has several statewide lottery options. One is the Pick 3 game in which you pick one of the 1000 three-digit numbers between 000 and 999\. The lottery selects a three-digit number at random. With a bet of \(\$ 1,\) you win \(\$ 500\) if your number is selected and nothing (\$0) otherwise. (Many states have a very similar type of lottery.) (Source: Background information from www.ohiolottery.com.) a. With a single \(\$ 1\) bet, what is the probability that you win \(\$ 500 ?\) b. Let \(X\) denote your winnings for a \(\$ 1\) bet, so \(x=\$ 0\) or \(x=\$ 500\). Construct the probability distribution for \(X\). c. Show that the mean of the distribution equals 0.50 , corresponding to an expected return of 50 cents for the dollar paid to play. Interpret the mean. d. In Ohio's Pick 4 lottery, you pick one of the 10,000 four-digit numbers between 0000 and 9999 and (with a \(\$ 1\) bet ) win \(\$ 5000\) if you get it correct. In terms of your expected winnings, with which game are you better off - playing Pick 4 , or playing Pick 3 in which you win \(\$ 500\) for a correct choice of a three-digit number? Justify your answer.

Binomial needs fixed \(n\) For the binomial distribution, the number of trials \(n\) is a fixed number. Let \(X\) denote the number of girls in a randomly selected family in Canada that has three children. Let \(Y\) denote the number of girls in a randomly selected family in Canada (that is, the number of children could be any number). A binomial distribution approximates well the probability distribution for one of \(X\) and \(Y,\) but not for the other. a. Explain why. b. Identify the case for which the binomial applies, and identify \(n\) and \(p\).

A World Health Organization study (the MONICA project) of health in various countries reported that in Canada, systolic blood pressure readings have a mean of 121 and a standard deviation of \(16 .\) A reading above 140 is considered to be high blood pressure. a. What is the \(z\) -score for a blood pressure reading of \(140 ?\) b. If systolic blood pressure in Canada has a normal distribution, what proportion of Canadians suffers from high blood pressure? c. What proportion of Canadians has systolic blood pressures in the range from 100 to \(140 ?\) d. Find the 90 th percentile of blood pressure readings.

The Internet site www.ItsJustLunch .com advertises itself as a dating service for busy professionals that has set up over two million first dates for lunch or drinks after work. An advertisement for this site stated that a survey of their users found that a woman has chance 1 in 8 of a second date if she has not heard from the man within 24 hours of their first date. On Saturday, Shawna had a luncheon date with Jack and a dinner date with Lawrence. By Sunday evening she had not heard from either of them. Based on the information claimed by www.ItsJustLunch.com, construct a table with the probability distribution of \(X=\) the number of these men 2) with whom she has a second date. (Source: \((0,1,\) or Background information from www.ItsJustLunch.com.)
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