8.2.3.3 - One Sample Mean Z Test (Optional) | STAT 200

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8.2.3.3 - One Sample Mean z Test (Optional)

A one sample mean \(z\) test is used when the population is known to be normally distributed and when the population standard deviation (\(\sigma\)) is known. This most frequently occurs in the social sciences when standardized measures are used such as IQ, SAT, ACT, or GRE scores, for which the population parameters are known.

The formula for computing a \(z\) test statistic for one sample mean is identical to that of computing a \(t\) test statistic for one sample mean, except now the population standard deviation is known and can be used in computing the standard error.

z Test Statistic: One Group Mean

\(z=\dfrac{\overline{x}-\mu_0}{\dfrac{\sigma}{\sqrt{n}}}\)

\(\overline{x}\) = sample mean\(\mu_{0}\) = hypothesized population mean\(s\) = sample standard deviation\(n\) = sample size

The other primary difference between the one sample mean \(t\) test and the one sample mean \(z\) test is the latter uses the standard normal distribution (i.e., \(z\) distribution) in determining the \(p\)-value. Below are the directions for conducting a one sample mean \(z\) test in Minitab.

Minitab® – Performing a One Sample Mean z Test

Research question: Are the IQ scores of students at one college-prep school above the national average?

Scores on one American IQ test are normed to have a mean of 100 and standard deviation of 15. In a simple random sample of 25 students at this school the mean was 110.

To perform a one-sample mean z test in Minitab using summarized data:

1. In Minitab, select Stat > Basic Statistics > 1-sample Z
2. Select Summarized data from the dropdown
3. Enter 25 for the sample size, 110 for the sample mean and 15 for the known standard deviation.
4. Check the box Perform a hypothesis test
5. For the Hypothesized mean enter 100
6. Select Options
7. Use the default Alternative hypothesis of Mean > hypothesized value
8. Use the default Confidence level of 95
9. Click OK and OK

This should result in the following output:

Descriptive Statistics

N	Mean	SE Mean	95% Lower Bound for \(\mu\)
25	110.00	3.00	105.07

\(\mu\): population mean of SampleKnown standard deviation = 15

Test

Null hypothesis	H0: \(\mu\) = 100
Alternative hypothesis	H1: \(\mu\) > 100

Z-Value	P-Value
3.33	0.000

Summary of Results Section

We could summarize these results using the five step hypothesis testing procedure:

1. Check assumptions and write hypotheses

The population is known to be normally distributed and the population standard deviation is known to be 15. With these two conditions met we can conduct a one sample mean z test

\(H_0\colon \mu = 100\)\(H_a\colon \mu > 100\)

2. Calculate the test statistic

From the Minitab output, \(z = 3.33\)

3. Determine the p-value

From the Minitab output, \(p = 0.000\)

4. Make a decision

\(p \le \alpha\), reject the null hypothesis

5. State a "real world" conclusion

There is convincing evidence that the mean IQ score of all students at this school is greater than 100.

• « Previous8.2.3.2.2 - Minitab: 1 Sample Mean t Test, Summarized Data
• Next8.3 - Paired Means »

Lessons

• Welcome to STAT 200!

• 0: Prerequisite Skills
  • 0.1 - Review of Algebra
    • 0.1.1 - Order of Operations
    • 0.1.2 - Summations
    • 0.1.3 - Basic Linear Equations
  • 0.2 - Introduction to Minitab
  • 0.3 - Word's Equation Editor
  • 0.4 - Canvas' Equation Editor

• 1: Collecting Data
  • 1.1 - Cases & Variables
    • 1.1.1 - Categorical & Quantitative Variables
    • 1.1.2 - Explanatory & Response Variables
  • 1.2 - Samples & Populations
    • 1.2.1 - Sampling Bias
    • 1.2.2 - Sampling Methods
      • 1.2.2.1 - Minitab: Simple Random Sampling
  • 1.3 - Other Sources of Bias
  • 1.4 - Research Study Design
    • 1.4.1 - Confounding Variables
    • 1.4.2 - Causal Conclusions
    • 1.4.3 - Independent and Paired Samples
    • 1.4.4 - Control and Placebo Groups
    • 1.4.5 - Blinding
  • 1.5 - Lesson 1 Summary

• 2: Describing Data, Part 1
  • 2.1 - Categorical Variables
    • 2.1.1 - One Categorical Variable
      • 2.1.1.1 - Risk and Odds
      • 2.1.1.2 - Visual Representations
        • 2.1.1.2.1 - Minitab: Frequency Tables
        • 2.1.1.2.2 - Minitab: Pie Charts
        • 2.1.1.2.3 - Minitab: Bar Charts
    • 2.1.2 - Two Categorical Variables
      • 2.1.2.1 - Minitab: Two-Way Contingency Table
      • 2.1.2.2 - Minitab: Clustered Bar Chart
      • 2.1.2.3 - Minitab: Stacked Bar Chart
    • 2.1.3 - Probability Rules
      • 2.1.3.1 - Range of Probabilities
      • 2.1.3.2 - Combinations of Events
        • 2.1.3.2.1 - Disjoint & Independent Events
        • 2.1.3.2.2 - Intersections
        • 2.1.3.2.3 - Unions
        • 2.1.3.2.4 - Complements
        • 2.1.3.2.5 - Conditional Probability
          • 2.1.3.2.5.1 - Advanced Conditional Probability Applications
  • 2.2 - One Quantitative Variable
    • 2.2.1 - Graphs: Dotplots and Histograms
    • 2.2.2 - Outliers
    • 2.2.3 - Shape
    • 2.2.4 - Measures of Central Tendency
      • 2.2.4.1 - Skewness & Central Tendency
    • 2.2.5 - Measures of Spread
    • 2.2.6 - Minitab: Central Tendency & Variability
    • 2.2.7 - The Empirical Rule
    • 2.2.8 - z-scores
    • 2.2.9 - Percentiles
    • 2.2.10 - Five Number Summary
  • 2.3 - Lesson 2 Summary

• 3: Describing Data, Part 2
  • 3.1 - Single Boxplot
  • 3.2 - Identifying Outliers: IQR Method
  • 3.3 - One Quantitative and One Categorical Variable
  • 3.4 - Two Quantitative Variables
    • 3.4.1 - Scatterplots
      • 3.4.1.1 - Minitab: Simple Scatterplot
    • 3.4.2 - Correlation
      • 3.4.2.1 - Formulas for Computing Pearson's r
      • 3.4.2.2 - Example of Computing r by Hand (Optional)
      • 3.4.2.3 - Minitab: Compute Pearson's r
    • 3.4.3 - Simple Linear Regression
      • 3.4.3.1 - Minitab: SLR
      • 3.4.3.2 - Example: Interpreting Output
  • 3.5 - Relations between Multiple Variables
    • 3.5.1 - Scatterplot with Groups
    • 3.5.2 - Bubble Plots
    • 3.5.3 - Time Series Plot
  • 3.6 - Lesson 3 Summary

• 4: Confidence Intervals
  • 4.1 - Sampling Distributions
    • 4.1.1 - StatKey Examples
      • 4.1.1.1 - NFL Salaries (One Mean)
      • 4.1.1.2 - Coin Flipping (One Proportion)
    • 4.1.2 - Copying Data into StatKey
    • 4.1.3 - Impact of Sample Size
  • 4.2 - Introduction to Confidence Intervals
    • 4.2.1 - Interpreting Confidence Intervals
    • 4.2.2 - Applying Confidence Intervals
  • 4.3 - Introduction to Bootstrapping
    • 4.3.1 - Example: Bootstrap Distribution for Proportion of Peanuts
    • 4.3.2 - Example: Bootstrap Distribution for Difference in Mean Exercise
  • 4.4 - Bootstrap Confidence Interval
    • 4.4.1 - StatKey: Standard Error Method
      • 4.4.1.1 - Example: Proportion of Lactose Intolerant German Adults
      • 4.4.1.2 - Example: Difference in Mean Commute Times
    • 4.4.2 - StatKey: Percentile Method
      • 4.4.2.1 - Example: Correlation Between Quiz & Exam Scores
      • 4.4.2.2 - Example: Difference in Dieting by Biological Sex
      • 4.4.2.3 - Example: One sample mean sodium content
  • 4.5 - Paired Samples
  • 4.6 - Impact of Sample Size on Confidence Intervals
  • 4.7 - Lesson 4 Summary

• 5: Hypothesis Testing, Part 1
  • 5.1 - Introduction to Hypothesis Testing
  • 5.2 - Writing Hypotheses
    • 5.2.1 - Examples
  • 5.3 - Randomization Procedures
    • 5.3.1 - StatKey Randomization Methods (Optional)
  • 5.4 - p-values
  • 5.5 - Randomization Test Examples in StatKey
    • 5.5.1 - Single Proportion Example: PA Residency
    • 5.5.2 - Paired Means Example: Age
    • 5.5.3 - Difference in Means Example: Exercise by Biological Sex
    • 5.5.4 - Correlation Example: Quiz & Exam Scores
  • 5.6 - Lesson 5 Summary

• 6: Hypothesis Testing, Part 2
  • 6.1 - Type I and Type II Errors
  • 6.2 - Significance Levels
  • 6.3 - Issues with Multiple Testing
  • 6.4 - Practical Significance
  • 6.5 - Power
  • 6.6 - Confidence Intervals & Hypothesis Testing
  • 6.7 - Lesson 6 Summary

• 7: Normal Distributions
  • 7.1 - Standard Normal Distribution
  • 7.2 - Minitab: Finding Proportions Under a Normal Distribution
    • 7.2.1 - Proportion 'Less Than'
      • 7.2.1.1 - Example: P(Z<-1)
      • 7.2.1.2 - Example: P(SATM<540)
    • 7.2.2 - Proportion 'Greater Than'
      • 7.2.2.1 - Example: P(Z>0.5)
    • 7.2.3 - Proportion 'In between'
      • 7.2.3.1 - Example: Proportion Between z -2 and +2
    • 7.2.4 - Proportion 'More Extreme Than'
  • 7.3 - Minitab: Finding Values Given Proportions
    • 7.3.1 - Top X%
    • 7.3.2 - Bottom X%
    • 7.3.3 - Middle X%
  • 7.4 - Central Limit Theorem
    • 7.4.1 - Hypothesis Testing
      • 7.4.1.1 - Video Example: Mean Body Temperature
      • 7.4.1.2 - Video Example: Correlation Between Printer Price and PPM
      • 7.4.1.3 - Example: Proportion NFL Coin Toss Wins
      • 7.4.1.4 - Example: Proportion of Women Students
      • 7.4.1.5 - Example: Mean Quiz Score
      • 7.4.1.6 - Example: Difference in Mean Commute Times
    • 7.4.2 - Confidence Intervals
      • 7.4.2.1 - Video Example: 98% CI for Mean Atlanta Commute Time
      • 7.4.2.2 - Video Example: 90% CI for the Correlation between Height and Weight
      • 7.4.2.3 - Example: 99% CI for Proportion of Women Students
  • 7.5 - Lesson 7 Summary

• 8: Inference for One Sample
  • 8.1 - One Sample Proportion
    • 8.1.1 - Confidence Intervals
      • 8.1.1.1 - Normal Approximation Formulas
        • 8.1.1.1.1 - Video Example: PA Residency
        • 8.1.1.1.2 - Video Example: Dog Ownership
        • 8.1.1.1.3 - Video Example: Books
        • 8.1.1.1.4 - Example: Retirement
      • 8.1.1.2 - Minitab: Confidence Interval for a Proportion
        • 8.1.1.2.1 - Example with Summarized Data
        • 8.1.1.2.2 - Example with Summarized Data
      • 8.1.1.3 - Computing Necessary Sample Size
    • 8.1.2 - Hypothesis Testing
      • 8.1.2.1 - Normal Approximation Method Formulas
        • 8.1.2.1.1 - Video Example: Male Babies
        • 8.1.2.1.2 - Example: Handedness
        • 8.1.2.1.3 - Example: Ice Cream
        • 8.1.2.1.4 - Example: Overweight Citizens
      • 8.1.2.2 - Minitab: Hypothesis Tests for One Proportion
        • 8.1.2.2.1 - Minitab: 1 Proportion z Test, Raw Data
        • 8.1.2.2.2 - Minitab: 1 Sample Proportion z test, Summary Data
          • 8.1.2.2.2.1 - Minitab Example: Normal Approx. Method
  • 8.2 - One Sample Mean
    • 8.2.1 - t Distribution
    • 8.2.2 - Confidence Intervals
      • 8.2.2.1 - Formulas
        • 8.2.2.1.1 - Example: MLB Age
        • 8.2.2.1.2- Example: Sleep Deprivation
        • 8.2.2.1.3 - Example: Milk
      • 8.2.2.2 - Minitab: Confidence Interval of a Mean
        • 8.2.2.2.1 - Example: Age of Pitchers (Summarized Data)
        • 8.2.2.2.2 - Example: Coffee Sales (Data in Column)
      • 8.2.2.3 - Computing Necessary Sample Size
        • 8.2.2.3.1 - Example: Estimating IQ
        • 8.2.2.3.2 - Video Example: Age
        • 8.2.2.3.3 - Video Example: Cookie Weights
    • 8.2.3 - Hypothesis Testing
      • 8.2.3.1 - One Sample Mean t Test, Formulas
        • 8.2.3.1.1 - Video Example: Book Costs
        • 8.2.3.1.2 : Example: Pulse Rate
        • 8.2.3.1.3 - Example: Coffee
        • 8.2.3.1.4 - Example: Transportation Costs
      • 8.2.3.2 - Minitab: One Sample Mean t Tests
        • 8.2.3.2.1 - Minitab: 1 Sample Mean t Test, Raw Data
        • 8.2.3.2.2 - Minitab: 1 Sample Mean t Test, Summarized Data
      • 8.2.3.3 - One Sample Mean z Test (Optional)
  • 8.3 - Paired Means
    • 8.3.1 - Confidence Intervals
      • 8.3.1.1. - Example: Change in Knowledge
      • 8.3.1.2 - Video Example: Difference in Exam Scores
    • 8.3.2 - Hypothesis Testing
      • 8.3.2.1 - Example: Quiz Scores
    • 8.3.3 - Minitab: Paired Means Test
      • 8.3.3.1 - Example: SAT Scores
      • 8.3.3.2 - Example: Marriage Age (Summarized Data)
  • 8.4 - Lesson 8 Summary

• 9: Inference for Two Samples
  • 9.1 - Two Independent Proportions
    • 9.1.1 - Confidence Intervals
      • 9.1.1.1 - Minitab: Confidence Interval for 2 Proportions
    • 9.1.2 - Hypothesis Testing
      • 9.1.2.1 - Normal Approximation Method Formulas
        • 9.1.2.1.1 – Example: Ice Cream
        • 9.1.2.1.2 – Example: Same Sex Marriage
      • 9.1.2.2 - Minitab: Difference Between 2 Independent Proportions
        • 9.1.2.2.1 - Example: Dating
  • 9.2 - Two Independent Means
    • 9.2.1 - Confidence Intervals
      • 9.2.1.1 - Minitab: Confidence Interval Between 2 Independent Means
        • 9.2.1.1.1 - Video Example: Mean Difference in Exam Scores, Summarized Data
    • 9.2.2 - Hypothesis Testing
      • 9.2.2.1 - Minitab: Independent Means t Test
        • 9.2.2.1.1 - Example: Summarized Data
        • 9.2.2.1.3 - Example: Height by Sex
  • 9.3 - Lesson 9 Summary

• 10: One-Way ANOVA
  • 10.1 - Introduction to the F Distribution
  • 10.2 - Hypothesis Testing
  • 10.3 - Pairwise Comparisons
  • 10.4 - Minitab: One-Way ANOVA
  • 10.5 - Example: SAT-Math Scores by Award Preference
  • 10.6 - Example: Exam Grade by Professor
  • 10.7 - Lesson 10 Summary

• 11: Chi-Square Tests
  • 11.1 - Reviews
    • 11.1.1 - Frequency Table
    • 11.1.2 - Two-Way Contingency Table
    • 11.1.3 - Probability Distribution Plots
    • 11.1.4 - Conditional Probabilities and Independence
  • 11.2 - Goodness of Fit Test
    • 11.2.1 - Five Step Hypothesis Testing Procedure
      • 11.2.1.1 - Video: Cupcakes (Equal Proportions)
      • 11.2.1.2- Cards (Equal Proportions)
      • 11.2.1.3 - Roulette Wheel (Different Proportions)
    • 11.2.2 - Minitab: Goodness-of-Fit Test
      • 11.2.2.1 - Example: Summarized Data, Equal Proportions
      • 11.2.2.2 - Example: Summarized Data, Different Proportions
  • 11.3 - Chi-Square Test of Independence
    • 11.3.1 - Example: Gender and Online Learning
    • 11.3.2 - Minitab: Test of Independence
      • 11.3.2.1 - Example: Raw Data
      • 11.3.2.2 - Example: Summarized Data
    • 11.3.3 - Relative Risk
  • 11.4 - Lesson 11 Summary

• 12: Correlation & Simple Linear Regression
  • 12.1 - Review: Scatterplots
  • 12.2 - Correlation
    • 12.2.1 - Hypothesis Testing
      • 12.2.1.1 - Example: Quiz & Exam Scores
      • 12.2.1.2 - Example: Age & Height
      • 12.2.1.3 - Example: Temperature & Coffee Sales
    • 12.2.2 - Correlation Matrix
      • 12.2.2.1 - Example: Student Survey
      • 12.2.2.2 - Example: Body Correlation Matrix
  • 12.3 - Simple Linear Regression
    • 12.3.1 - Formulas
    • 12.3.2 - Assumptions
    • 12.3.3 - Minitab - Simple Linear Regression
    • 12.3.4 - Hypothesis Testing for Slope
      • 12.3.4.1 - Example: Quiz and exam scores
      • 12.3.4.2 - Example: Business Decisions
    • 12.3.5 - Confidence Interval for Slope
      • 12.3.5.1 - Example: Quiz and exam scores
  • 12.4 - Coefficient of Determination
  • 12.5 - Cautions
  • 12.6 - Correlation & Regression Example
  • 12.7 - Lesson 12 Summary

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