Table of contents

1. Intro to Stats and Collecting Data1h 14m
2. Describing Data with Tables and Graphs1h 55m
3. Describing Data Numerically2h 5m
4. Probability2h 16m
5. Binomial Distribution & Discrete Random Variables3h 6m
6. Normal Distribution and Continuous Random Variables2h 11m
7. Sampling Distributions & Confidence Intervals: Mean3h 23m
8. Sampling Distributions & Confidence Intervals: Proportion1h 25m
9. Hypothesis Testing for One Sample3h 29m
10. Hypothesis Testing for Two Samples4h 50m
11. Correlation1h 6m
12. Regression1h 50m
13. Chi-Square Tests & Goodness of Fit1h 57m
14. ANOVA1h 57m

3. Describing Data Numerically

Standard Deviation

Statistics

3. Describing Data Numerically

Standard Deviation

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2:54 minutes

Problem 2.Q.6a

Textbook Question

Refer to the sample statistics from Exercise 5 and determine whether any of the house prices below are unusual. Explain your reasoning.

a. $225,000

Verified step by step guidance

Step 1: Recall the rule for identifying unusual values. A value is considered unusual if it lies more than 2 standard deviations away from the mean. Mathematically, this can be expressed as: $ \text{Unusual if: } x < \mu - 2\sigma \text{ or } x > \mu + 2\sigma $, where $ \mu $ is the mean and $ \sigma $ is the standard deviation.

Step 2: Obtain the mean ($ \mu $) and standard deviation ($ \sigma $) of the house prices from the sample statistics provided in Exercise 5. These values are necessary to calculate the range of usual values.

Step 3: Calculate the lower bound of usual values using the formula $ \mu - 2\sigma $. Substitute the mean and standard deviation into the formula to find this value.

Step 4: Calculate the upper bound of usual values using the formula $ \mu + 2\sigma $. Again, substitute the mean and standard deviation into the formula to find this value.

Step 5: Compare the given house price of $225,000 to the calculated range of usual values. If $225,000 lies outside the range $ [\mu - 2\sigma, \mu + 2\sigma] $, it is considered unusual. Otherwise, it is not unusual.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Unusual Values

In statistics, an unusual value, or outlier, is a data point that significantly differs from the other observations in a dataset. Typically, values that lie beyond 1.5 times the interquartile range (IQR) from the quartiles are considered unusual. Identifying unusual values helps in understanding the distribution and variability of the data.

Descriptive Statistics

Descriptive statistics summarize and describe the main features of a dataset. Key measures include the mean, median, mode, range, variance, and standard deviation. These statistics provide a foundation for understanding the overall trends and patterns in the data, which is essential for determining whether specific values are unusual.

Normal Distribution

A normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. Understanding the properties of normal distribution, including the empirical rule (68-95-99.7), helps in assessing how likely a particular value is to be considered unusual within the context of the dataset.

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Related Practice

Textbook Question

Constructing Data Sets In Exercises 25–28, construct a data set that has the given statistics.

n = 6

x̄ = 7

s ≈ 2

Textbook Question

In Exercises 27 and 28, find the range, mean, variance, and standard deviation of the sample data set.

Salaries (in dollars) of a random sample of teachers

62,222 56,719 50,259 45,120 47,692 45,985 53,489 71,534

Textbook Question

The mean sale per customer for 40 customers at a gas station is $32.00, with a standard deviation of $4.00. Using Chebychev’s Theorem, determine at least how many of the customers spent between $24.00 and $40.00.

Textbook Question

From a random sample of airplanes, the number of defects found in their fuselages are listed. Find the sample mean and the sample standard deviation of the data.

Textbook Question

Refer to the sample statistics from Exercise 5 and determine whether any of the house prices below are unusual. Explain your reasoning.

d. $147,000

Textbook Question

Using the Empirical Rule In Exercises 29–34, use the Empirical Rule.

Use the sample statistics from Exercise 29 and assume the number of vehicles in the sample is 75.

a. Estimate the number of vehicles whose speeds are between 63 miles per hour and 71 miles per hour.

Textbook Question

Use frequency distribution formulas to estimate the sample mean and the sample standard deviation of the data set in Exercise 2.

Textbook Question

Shifting Data Sample annual salaries (in thousands of dollars) for employees at a company are listed.

40 35 49 53 38 39 40

37 49 34 38 43 47 35

c. Each employee in the sample takes a pay cut of $2000 from their original salary. Find the sample mean and the sample standard deviation for the revised data set.

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