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Chapter 6: Problem 54

College women have heights with the following distribution (inches): \(N(65,2.5)\). a. Find the height at the 75 th percentile. b. Find the height at the 25 th percentile. c. Find the interquartile range for heights. d. Is the interquartile range larger or smaller than the standard deviation? Explain.

Short Answer

Expert verified
The height at the 75th percentile is approximately \(65 + 0.6745*2.5 = 66.69\) inches. The height at the 25th percentile is approximately \(65 + (-0.6745)*2.5 = 63.31\) inches. The interquartile range for the heights is roughly \(66.69 - 63.31 = 3.37\) inches. Therefore, the interquartile range is larger than the standard deviation of 2.5 inches.

Step by step solution

01

Find the height at the 75th percentile

The 75th percentile is known as the third quartile (Q3), and for a normal distribution we can use the z-value corresponding to a cumulative probability of 0.75. We interpret the Z score from standard normal distribution tables which gives us roughly 0.6745. Using formula \(X = \mu + Z\sigma\) where \(\mu\) is the mean and \(\sigma\) is the standard deviation and Z is the value obtained from the table, we substitute the values to get \(X = 65 + 0.6745*2.5\).
02

Find the height at the 25th percentile

Similarly, the 25th percentile is the first quartile (Q1). The z-value for this cumulative probability of 0.25 is -0.6745 (remember, the z-table gives values for below the mean). Using the same formula \(X = \mu + Z\sigma\) we calculate \(X = 65 + (-0.6745)*2.5\).
03

Find the interquartile range

The interquartile range (IQR) is the range in which the middle 50% of the values fall, and is calculated as the difference between the third quartile (Q3) and the first quartile (Q1). In short, it is the difference between the height at the 75th percentile and the height at the 25th percentile.
04

Compare the interquartile range with standard deviation

Take the interquartile range calculated in step 3 and compare it with the standard deviation (2.5 inches in this case). Determine if the range is larger or smaller.

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Key Concepts

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

Understanding Percentiles in a Normal Distribution
A percentile is a measure used in statistics to indicate the value below which a given percentage of observations fall. For example, the 75th percentile is the value below which 75% of the data can be found.

In a normal distribution, you can determine percentiles by using the formula \(X = \mu + Z\sigma\), where \(\mu\) is the mean, \(\sigma\) is the standard deviation, and \(Z\) is the z-value from the standard normal distribution table.
  • For the 75th percentile (also known as the third quartile, \(Q3\)), the z-value is approximately 0.6745. You calculate the height by substituting into the formula: \(X = 65 + 0.6745 \times 2.5\).
  • For the 25th percentile (the first quartile, \(Q1\)), the z-value is -0.6745. For this percentile: \(X = 65 + (-0.6745) \times 2.5\).
These calculations involve looking up the z-value in a standard normal distribution table, which is a tool giving percentages of data points expected below a given z-score.
What is Standard Deviation?
Standard deviation is a statistic that measures the spread or variability of a set of data values. In simpler terms, it tells you how much the individual data points typically deviate from the mean of the dataset.

For instance, in our example with heights of college women, the normal distribution is described by \(N(65, 2.5)\):
  • The number 65 is the mean height.
  • The number 2.5 is the standard deviation, which reflects how spread out the heights are around the mean.
A smaller standard deviation would mean the heights are closely clustered around the mean. A larger one implies more variability. In general, around 68% of data points fall within one standard deviation from the mean in a normal distribution.
Exploring the Interquartile Range (IQR)
The Interquartile Range (IQR) is a measure of statistical dispersion, or how spread out the values are in a dataset. It specifically looks at the middle 50% of the data distribution.

To find the IQR, subtract the first quartile (\(Q1\)) from the third quartile (\(Q3\)). In mathematical terms, \(IQR = Q3 - Q1\).
  • For the heights example, these quartiles represent the 25th and 75th percentiles.
  • The IQR encompasses the range in which the central half of the data fall, providing a more robust measure than the entire range as it's less affected by extreme outliers.
When comparing IQR with standard deviation, you might find varied results. For normally distributed data, IQR is often larger than a single standard deviation. This means the IQR can give a more stable reflection of variability as it considers only the central data points.

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Most popular questions from this chapter

The distribution of the math portion of SAT scores has a mean of 500 and a standard deviation of 100 , and the scores are approximately Normally distributed. a. What is the probability that one randomly selected person will have an SAT score of 550 or more? b. What is the probability that four randomly selected people will all have SAT scores of 550 or more? c. For 800 randomly selected people, what is the probability that 250 or more will have scores of 550 or more? d. For 800 randomly selected people, on average how many should have scores of 550 or more? Round to the nearest whole number. e. Find the standard deviation for part d. Round to the nearest whole number. f. Report the range of people out of 800 who should have scores of 550 or more from two standard deviations below the mean to two standard deviations above the mean. Use your rounded answers to part d and e. g. If 400 out of 800 randomly selected people had scores of 550 or more, would you be surprised? Explain.

Use technology or a Normal table to find each of the following. Include an appropriately labeled sketch of the Normal curve for each part with the appropriate area shaded. a. Find the probability that a \(z\) -score will be \(2.12\) or greater. b. Find the probability that a \(z\) -score will be less than \(-0.74\). c. Find the probability that a \(z\) -score will between \(1.25\) and \(2.37\).

Toronto drivers have been going to small towns in Ontario (Canada) to take the drivers' road test, rather than taking the test in Toronto, because the pass rate in the small towns is \(90 \%\), which is much higher than the pass rate in Toronto. Suppose that every day, 100 people independently take the test in one of these small towns. a. What is the number of people who are expected to pass? b. What is the standard deviation for the number expected to pass? c. After a great many days, according to the Empirical Rule, on about \(95 \%\) of these days the number of people passing the test will be as low as and as high as d. If you found that on one day, 89 out of 100 passed the test, would you consider this to be a very high number?

Assume college women's heights are approximately Normally distributed with a mean of 65 inches and a standard deviation of \(2.5\) inches. Choose the Stat- \(-\) Crunch output for finding the percentage of college women who are taller than 67 inches and report the correct percentage. Round to one decimal place. a. b.

A coin will be flipped four times, and the number of heads recorded. Explain why this is a binomial experiment. Check all four required conditions.
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