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Chapter 6: Problem 30

For an approximately normally distributed random variable \(X\) with a mean of 200 and a standard deviation of 36 , a. Find the \(z\) -score corresponding to the lower quartile and upper quartile of the standard normal distribution. b. Find and interpret the lower quartile and upper quartile of \(X\). c. Find the interquartile range (IQR) of \(X\). d. An observation is a potential outlier if it is more than \(1.5 \times\) IQR below \(\mathrm{Q} 1\) or above \(\mathrm{Q} 3\). Find the values of \(X\) that would be considered potential outliers.

Short Answer

Expert verified
Potential outliers are below 102.8 or above 297.2 for X.

Step by step solution

01

Understanding Quartiles on a Standard Normal Distribution

For a standard normal distribution, the lower quartile (Q1) is at the 25th percentile, and the upper quartile (Q3) is at the 75th percentile. The z-scores corresponding to these percentiles are approximately -0.675 and 0.675.
02

Calculate Lower and Upper Quartiles for X

To find the lower and upper quartiles for X, use the z-score formula: \( z = \frac{(X - \text{mean})}{\text{standard deviation}} \). For Q1, \( X_1 = 200 + (-0.675 \times 36) = 175.7 \). For Q3, \( X_3 = 200 + (0.675 \times 36) = 224.3 \). These values represent the 25th and 75th percentiles of X.
03

Calculate the Interquartile Range (IQR) of X

The IQR is the difference between the upper quartile and the lower quartile. Calculate it using: \( IQR = 224.3 - 175.7 = 48.6 \).
04

Determine Potential Outliers

An observation is a potential outlier if it is below \( Q1 - 1.5 \times IQR \) or above \( Q3 + 1.5 \times IQR \). Calculate: \( Q1 - 1.5 \times IQR = 175.7 - 1.5 \times 48.6 = 102.8 \), and \( Q3 + 1.5 \times IQR = 224.3 + 1.5 \times 48.6 = 297.2 \). Thus, potential outliers are observations less than 102.8 or greater than 297.2.

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Key Concepts

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

Quartiles
Quartiles are statistical measures that help us divide a dataset into four equal parts. This division offers insights into the distribution of the data. In the context of a normally distributed random variable, quartiles are particularly useful because they help describe the data's spread and central tendency.

- **Lower Quartile (Q1):** This is the 25th percentile of a dataset. It means that 25% of the data falls below this value. - **Upper Quartile (Q3):** This is the 75th percentile, indicating that 75% of the data falls below this point.

For a standard normal distribution, Q1 and Q3 correspond to z-scores of approximately -0.675 and 0.675, respectively. When these z-scores are translated back to the original scale using the mean and standard deviation, they give the quartile values for the dataset. This method helps interpret where a particular observation stands within the overall distribution.
Interquartile Range
The Interquartile Range (IQR) is a measure of statistical dispersion, or variability, based on dividing a dataset into quartiles. It is calculated as the difference between the upper and lower quartiles (Q3 - Q1).

The IQR is essential because:
  • It measures the middle 50% of the data, providing a robust summary that is not easily distorted by outliers or extreme values.
  • Being independent of extreme values means it's useful for understanding the spread of data.
To find the IQR, subtract the value of the lower quartile from the upper quartile. For example, if Q1 is 175.7 and Q3 is 224.3, then the IQR is 48.6. This value helps us understand how spread out the central portion of your data is.
Z-Score
A z-score, or standard score, indicates how many standard deviations an element is from the mean. In the realm of normal distributions, the z-score is crucial for determining the position of a score within the distribution.

- **Why Use Z-Scores?**
  • Z-scores enable the comparison of results from different datasets by standardizing the scores.
  • They help in identifying how far away a particular observation is from the mean, and whether it is above or below the average.
- **Calculating Z-Scores:** The formula for calculating a z-score is: \[z = \frac{(X - \text{mean})}{\text{standard deviation}} \]This formula helps you determine where a particular data point stands in relation to an entire dataset.
Z-scores are particularly useful in the determination of quartile positions. By understanding which z-scores equate to certain percentiles, you can interpret how a single data point relates to the overall distribution. This is key in assessing probabilities and making informed statistical decisions.

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Most popular questions from this chapter

An exit poll is taken of 3000 voters in a statewide election. Let \(X\) denote the number who voted in favor of a special proposition designed to lower property taxes and raise the sales tax. Suppose that in the population, exactly \(50 \%\) voted for it. a. Explain why this scenario would seem to satisfy the three conditions needed to use the binomial distribution. Identify \(n\) and \(p\) for the binomial. b. Find the mean and standard deviation of the probability distribution of \(X\). c. Using the normal distribution approximation, give an interval in which you would expect \(X\) almost certainly to fall, if truly \(p=0.50 .\) (Hint: You can follow the reasoning of Example 15 on racial profiling.) d. Now, suppose that the exit poll had \(x=1706 .\) What would this suggest to you about the actual value of \(p ?\)

Checking guidelines In a village having more than 100 adults, eight are randomly selected in order to form a committee of residents. \(40 \%\) of adults in the village are connected with agriculture. a. Verify that the guidelines have been satisfied about the relative sizes of the population and the sample, thus allowing the use of a binomial probability distribution for the number of selected adults connected with agriculture. b. Check whether the guideline was satisfied for this binomial distribution to be reasonably approximated by a normal distribution.

For each of the following situations, explain whether the binomial distribution applies for \(X\). a. You are bidding on four items available on eBay. You think that you will win the first bid with probability \(25 \%\) and the second through fourth bids with probability \(30 \%\). Let \(X\) denote the number of winning bids out of the four items you bid on. b. You are bidding on four items available on eBay. Each bid is for \(\$ 70\), and you think there is a \(25 \%\) chance of winning a bid, with bids being independent events. Let \(X\) be the total amount of money you pay for your winning bids.

You are bidding on four items available on eBay. You think that for each bid, you have a \(25 \%\) chance of winning it, and the outcomes of the four bids are independent events. Let \(X\) denote the number of winning bids out of the four items you bid on. a. Explain why the distribution of \(X\) can be modeled by the binomial distribution. b. Find the probability that you win exactly 2 bids. c. Find the probability that you win 2 bids or fewer. d. Find the probability that you win more than 2 bids.

Verify the empirical rule by using Table A, software, or a calculator to show that for a normal distribution, the probability (rounded to two decimal places) within a. 1 standard deviation of the mean equals 0.68 . b. 2 standard deviations of the mean equals 0.95 . c. 3 standard deviations of the mean is very close to 1.00 .
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