Table of contents

1. Intro to Stats and Collecting Data1h 14m

Intro to Stats
24m

Levels of Measurement
18m

Intro to Collecting Data
8m

Sampling Methods
23m

2. Describing Data with Tables and Graphs1h 56m

Visualizing Qualitative vs. Quantitative Data
4m

Frequency Distributions
35m

Histograms
14m

Bar Graphs and Pareto Charts
11m

Pie Charts
8m

Frequency Polygons
10m

Dot Plots
6m

Stemplots (Stem-and-Leaf Plots)
14m

Time-Series Graph
9m

3. Describing Data Numerically2h 5m

Mean
9m

Median
17m

Mode
7m

Standard Deviation
16m

Interpreting Standard Deviation
20m

Percentiles & Quartiles
14m

Describing Data Numerically Using a Graphing CalculatorBonus
10m

Boxplots
8m

Descriptive Statistics-ExcelBonus
11m

Boxplots-ExcelBonus
8m

4. Probability2h 17m

Basic Concepts of Probability
7m

Complements
6m

Addition Rule
17m

Multiplication Rule: Independent Events
11m

Introduction to Contingency Tables
17m

Multiplication Rule: Dependent Events
17m

Bayes' Theorem
13m

Fundamental Counting Principle
8m

Counting
37m

5. Binomial Distribution & Discrete Random Variables3h 6m

Discrete Random Variables
31m

Binomial Distribution
1h 7m

Finding Binomial Probabilities-ExcelBonus
17m

Poisson Distribution
40m

Finding Poisson Probabilities-ExcelBonus
15m

Hypergeometric Distribution
14m

6. Normal Distribution and Continuous Random Variables2h 11m

Uniform Distribution
18m

Standard Normal Distribution
39m

Probabilities & Z-Scores w/ Graphing CalculatorBonus
19m

Non-Standard Normal Distribution
21m

Finding Probabilities, Z Values, and X Values with the Normal Distribution-ExcelBonus
32m

7. Sampling Distributions & Confidence Intervals: Mean3h 23m

Sampling Distribution of the Sample Mean and Central Limit Theorem
19m

Distribution of Sample Mean - ExcelBonus
23m

Introduction to Confidence Intervals
15m

Confidence Intervals for Population Mean
1h 18m

Determining the Minimum Sample Size Required
12m

Finding Probabilities and T Critical Values - ExcelBonus
28m

Confidence Intervals for Population Means - ExcelBonus
25m

8. Sampling Distributions & Confidence Intervals: Proportion2h 10m

Sampling Distribution of Sample Proportion
29m

Confidence Intervals for Population Proportion
42m

Confidence Intervals for Population Proportion - ExcelBonus
12m

Chi Square Distribution
20m

Confidence Intervals for Population Variance
24m

9. Hypothesis Testing for One Sample5h 8m

Steps in Hypothesis Testing
1h 6m

Performing Hypothesis Tests: Means
1h 4m

Hypothesis Testing: Means - ExcelBonus
42m

Performing Hypothesis Tests: Proportions
37m

Hypothesis Testing: Proportions - ExcelBonus
27m

Performing Hypothesis Tests: Variance
12m

Critical Values and Rejection Regions
28m

Link Between Confidence Intervals and Hypothesis Testing
12m

Type I & Type II Errors
16m

10. Hypothesis Testing for Two Samples5h 37m

Two Proportions
1h 13m

Two Proportions Hypothesis Test - ExcelBonus
28m

Two Means - Unknown, Unequal Variance
1h 3m

Two Means - Unknown Variances Hypothesis Test - ExcelBonus
12m

Two Means - Unknown, Equal Variance
15m

Two Means - Unknown, Equal Variances Hypothesis Test - ExcelBonus
9m

Two Means - Known Variance
12m

Two Means - Sigma Known Hypothesis Test - ExcelBonus
21m

Two Means - Matched Pairs (Dependent Samples)
42m

Matched Pairs Hypothesis Test - ExcelBonus
12m

Two Variances and F Distribution
29m

Two Variances - Graphing CalculatorBonus
16m

11. Correlation1h 24m

Scatterplots & Intro to Correlation
26m

Correlation Coefficient
21m

Creating Scatterplots and FInding Correlation Coefficient - ExcelBonus
6m

Hypothesis Tests for Correlation Coefficient Using TI-84 Bonus
17m

Inferences for the Correlation Coefficient - ExcelBonus
11m

12. Regression3h 33m

Linear Regression & Least Squares Method
26m

Residuals
12m

Coefficient of Determination
12m

Regression Line Equation and Coefficient of Determination - ExcelBonus
8m

Finding Residuals and Creating Residual Plots - ExcelBonus
11m

Inferences for Slope
31m

Enabling Data Analysis ToolpakBonus
1m

Regression Readout of the Data Analysis Toolpak - ExcelBonus
21m

Prediction Intervals
13m

Prediction Intervals - ExcelBonus
19m

Multiple Regression - ExcelBonus
29m

Quadratic Regression
15m

Quadratic Regression - ExcelBonus
10m

13. Chi-Square Tests & Goodness of Fit2h 21m

Goodness of Fit Test
41m

Goodness of FIt Test Using TI-84Bonus
17m

Goodness of Fit Test - ExcelBonus
10m

Contingency Tables
12m

Independence Tests
14m

Homogeneity Tests
11m

Using Matrices on a TI-84Bonus
6m

Independence Test Using TI-84Bonus
12m

Independence Tests - ExcelBonus
13m

14. ANOVA2h 29m

Introduction to ANOVA
31m

One-Way ANOVA - ExcelBonus
12m

Multiple Comparisons: Tukey Test
14m

Multiple Comparisons: Tukey-Kramer Test
15m

Multiple Comparisons: Bonferoni Test
24m

Two-Way ANOVA
32m

Two-Way ANOVA - ExcelBonus
18m

7. Sampling Distributions & Confidence Intervals: Mean

Confidence Intervals for Population Mean

Statistics

7. Sampling Distributions & Confidence Intervals: Mean

Confidence Intervals for Population Mean

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1:37 minutes

Problem 6.Q.1a

Textbook Question

[APPLET] The winning times (in hours) for a sample of 20 randomly selected Boston Marathon Women’s Open Division champions from 1980 to 2019 are shown in the table at the left. Assume the population standard deviation is 0.068 hour. (Source: Boston Athletic Association)

a. Find the point estimate of the population mean.

Verified step by step guidance

Step 1: Understand the problem. The goal is to find the point estimate of the population mean, which is the sample mean. The sample mean is calculated by summing all the data points and dividing by the total number of data points.

Step 2: Identify the data provided. The table contains 20 winning times (in hours) for the Boston Marathon Women’s Open Division champions. These values are the sample data.

Step 3: Write the formula for the sample mean. The formula is:

x̄ = \frac{\sum_{i = 1}^{n} x_{i}}{n}

, where

x̄

is the sample mean,

n

is the number of data points, and

x_{i}

are the individual data points.

Step 4: Add all the values in the table. Carefully sum each of the 20 winning times provided in the table. This will give the total sum of the data points.

Step 5: Divide the total sum by the number of data points (20). This will yield the sample mean, which is the point estimate of the population mean.

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Key Concepts

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

Point Estimate

A point estimate is a single value used to estimate a population parameter. In this context, the point estimate of the population mean is calculated by taking the average of the winning times of the sampled Boston Marathon champions. It provides a quick summary of the data and serves as a best guess for the true population mean.

Population Mean

The population mean is the average of all values in a population, representing a central tendency. It is a key parameter in statistics, often denoted by the symbol μ. In this case, the population mean would reflect the average winning time of all Boston Marathon Women’s Open Division champions from 1980 to 2019, which we aim to estimate using the sample data.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. It indicates how much individual data points differ from the mean. In this question, the population standard deviation is given as 0.068 hours, which helps assess the reliability of the point estimate and the spread of winning times among the sampled champions.

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Textbook Question

When all other quantities remain the same, how does the indicated change affect the width of a confidence interval? Explain.

b. Increase in the sample size

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Textbook Question

When all other quantities remain the same, how does the indicated change affect the width of a confidence interval? Explain.

c. Increase in the population standard deviation

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Textbook Question

Determining a Minimum Sample Size Determine the minimum sample size required when you want to be 99% confident that the sample mean is within two units of the population mean and σ = 1.4. Assume the population is normally distributed.

140

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Textbook Question

Paint Can Volumes A paint manufacturer uses a machine to fill gallon cans with paint (see figure). The manufacturer wants to estimate the mean volume of paint the machine is putting in the cans within 0.5 ounce. Assume the population of volumes is normally distributed.

a. Determine the minimum sample size required to construct a 90% confidence interval for the population mean. Assume the population standard deviation is 0.75 ounce.

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Textbook Question

[APPLET] The winning times (in hours) for a sample of 20 randomly selected Boston Marathon Women’s Open Division champions from 1980 to 2019 are shown in the table at the left. Assume the population standard deviation is 0.068 hour. (Source: Boston Athletic Association)

b. Find the margin of error for a 95% confidence level.

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Textbook Question

d. Does it seem likely that the population mean could be greater than 2.52 hours? Explain.

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Textbook Question

You wish to estimate the mean winning time for Boston Marathon Women’s Open Division champions. The estimate must be within 2 minutes of the population mean. Determine the minimum sample size required to construct a 99% confidence interval for the population mean. Use the population standard deviation from Exercise 1.

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Textbook Question

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c. Repeat part (b), assuming σ = 3.5 minutes. Compare the results.

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