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Introductory Statistics

10.1 Two Population Means with Unknown Standard Deviations

Introductory Statistics10.1 Two Population Means with Unknown Standard Deviations
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  1. Preface
  2. 1 Sampling and Data
    1. Introduction
    2. 1.1 Definitions of Statistics, Probability, and Key Terms
    3. 1.2 Data, Sampling, and Variation in Data and Sampling
    4. 1.3 Frequency, Frequency Tables, and Levels of Measurement
    5. 1.4 Experimental Design and Ethics
    6. 1.5 Data Collection Experiment
    7. 1.6 Sampling Experiment
    8. Key Terms
    9. Chapter Review
    10. Practice
    11. Homework
    12. Bringing It Together: Homework
    13. References
    14. Solutions
  3. 2 Descriptive Statistics
    1. Introduction
    2. 2.1 Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs
    3. 2.2 Histograms, Frequency Polygons, and Time Series Graphs
    4. 2.3 Measures of the Location of the Data
    5. 2.4 Box Plots
    6. 2.5 Measures of the Center of the Data
    7. 2.6 Skewness and the Mean, Median, and Mode
    8. 2.7 Measures of the Spread of the Data
    9. 2.8 Descriptive Statistics
    10. Key Terms
    11. Chapter Review
    12. Formula Review
    13. Practice
    14. Homework
    15. Bringing It Together: Homework
    16. References
    17. Solutions
  4. 3 Probability Topics
    1. Introduction
    2. 3.1 Terminology
    3. 3.2 Independent and Mutually Exclusive Events
    4. 3.3 Two Basic Rules of Probability
    5. 3.4 Contingency Tables
    6. 3.5 Tree and Venn Diagrams
    7. 3.6 Probability Topics
    8. Key Terms
    9. Chapter Review
    10. Formula Review
    11. Practice
    12. Bringing It Together: Practice
    13. Homework
    14. Bringing It Together: Homework
    15. References
    16. Solutions
  5. 4 Discrete Random Variables
    1. Introduction
    2. 4.1 Probability Distribution Function (PDF) for a Discrete Random Variable
    3. 4.2 Mean or Expected Value and Standard Deviation
    4. 4.3 Binomial Distribution
    5. 4.4 Geometric Distribution
    6. 4.5 Hypergeometric Distribution
    7. 4.6 Poisson Distribution
    8. 4.7 Discrete Distribution (Playing Card Experiment)
    9. 4.8 Discrete Distribution (Lucky Dice Experiment)
    10. Key Terms
    11. Chapter Review
    12. Formula Review
    13. Practice
    14. Homework
    15. References
    16. Solutions
  6. 5 Continuous Random Variables
    1. Introduction
    2. 5.1 Continuous Probability Functions
    3. 5.2 The Uniform Distribution
    4. 5.3 The Exponential Distribution
    5. 5.4 Continuous Distribution
    6. Key Terms
    7. Chapter Review
    8. Formula Review
    9. Practice
    10. Homework
    11. References
    12. Solutions
  7. 6 The Normal Distribution
    1. Introduction
    2. 6.1 The Standard Normal Distribution
    3. 6.2 Using the Normal Distribution
    4. 6.3 Normal Distribution (Lap Times)
    5. 6.4 Normal Distribution (Pinkie Length)
    6. Key Terms
    7. Chapter Review
    8. Formula Review
    9. Practice
    10. Homework
    11. References
    12. Solutions
  8. 7 The Central Limit Theorem
    1. Introduction
    2. 7.1 The Central Limit Theorem for Sample Means (Averages)
    3. 7.2 The Central Limit Theorem for Sums
    4. 7.3 Using the Central Limit Theorem
    5. 7.4 Central Limit Theorem (Pocket Change)
    6. 7.5 Central Limit Theorem (Cookie Recipes)
    7. Key Terms
    8. Chapter Review
    9. Formula Review
    10. Practice
    11. Homework
    12. References
    13. Solutions
  9. 8 Confidence Intervals
    1. Introduction
    2. 8.1 A Single Population Mean using the Normal Distribution
    3. 8.2 A Single Population Mean using the Student t Distribution
    4. 8.3 A Population Proportion
    5. 8.4 Confidence Interval (Home Costs)
    6. 8.5 Confidence Interval (Place of Birth)
    7. 8.6 Confidence Interval (Women's Heights)
    8. Key Terms
    9. Chapter Review
    10. Formula Review
    11. Practice
    12. Homework
    13. References
    14. Solutions
  10. 9 Hypothesis Testing with One Sample
    1. Introduction
    2. 9.1 Null and Alternative Hypotheses
    3. 9.2 Outcomes and the Type I and Type II Errors
    4. 9.3 Distribution Needed for Hypothesis Testing
    5. 9.4 Rare Events, the Sample, Decision and Conclusion
    6. 9.5 Additional Information and Full Hypothesis Test Examples
    7. 9.6 Hypothesis Testing of a Single Mean and Single Proportion
    8. Key Terms
    9. Chapter Review
    10. Formula Review
    11. Practice
    12. Homework
    13. References
    14. Solutions
  11. 10 Hypothesis Testing with Two Samples
    1. Introduction
    2. 10.1 Two Population Means with Unknown Standard Deviations
    3. 10.2 Two Population Means with Known Standard Deviations
    4. 10.3 Comparing Two Independent Population Proportions
    5. 10.4 Matched or Paired Samples
    6. 10.5 Hypothesis Testing for Two Means and Two Proportions
    7. Key Terms
    8. Chapter Review
    9. Formula Review
    10. Practice
    11. Homework
    12. Bringing It Together: Homework
    13. References
    14. Solutions
  12. 11 The Chi-Square Distribution
    1. Introduction
    2. 11.1 Facts About the Chi-Square Distribution
    3. 11.2 Goodness-of-Fit Test
    4. 11.3 Test of Independence
    5. 11.4 Test for Homogeneity
    6. 11.5 Comparison of the Chi-Square Tests
    7. 11.6 Test of a Single Variance
    8. 11.7 Lab 1: Chi-Square Goodness-of-Fit
    9. 11.8 Lab 2: Chi-Square Test of Independence
    10. Key Terms
    11. Chapter Review
    12. Formula Review
    13. Practice
    14. Homework
    15. Bringing It Together: Homework
    16. References
    17. Solutions
  13. 12 Linear Regression and Correlation
    1. Introduction
    2. 12.1 Linear Equations
    3. 12.2 Scatter Plots
    4. 12.3 The Regression Equation
    5. 12.4 Testing the Significance of the Correlation Coefficient
    6. 12.5 Prediction
    7. 12.6 Outliers
    8. 12.7 Regression (Distance from School)
    9. 12.8 Regression (Textbook Cost)
    10. 12.9 Regression (Fuel Efficiency)
    11. Key Terms
    12. Chapter Review
    13. Formula Review
    14. Practice
    15. Homework
    16. Bringing It Together: Homework
    17. References
    18. Solutions
  14. 13 F Distribution and One-Way ANOVA
    1. Introduction
    2. 13.1 One-Way ANOVA
    3. 13.2 The F Distribution and the F-Ratio
    4. 13.3 Facts About the F Distribution
    5. 13.4 Test of Two Variances
    6. 13.5 Lab: One-Way ANOVA
    7. Key Terms
    8. Chapter Review
    9. Formula Review
    10. Practice
    11. Homework
    12. References
    13. Solutions
  15. A | Review Exercises (Ch 3-13)
  16. B | Practice Tests (1-4) and Final Exams
  17. C | Data Sets
  18. D | Group and Partner Projects
  19. E | Solution Sheets
  20. F | Mathematical Phrases, Symbols, and Formulas
  21. G | Notes for the TI-83, 83+, 84, 84+ Calculators
  22. H | Tables
  23. Index
  1. The two independent samples are simple random samples from two distinct populations.
  2. For the two distinct populations:
    • if the sample sizes are small, the distributions are important (should be normal)
    • if the sample sizes are large, the distributions are not important (need not be normal)

NOTE

The test comparing two independent population means with unknown and possibly unequal population standard deviations is called the Aspin-Welch t-test. The degrees of freedom formula was developed by Aspin-Welch.

The comparison of two population means is very common. A difference between the two samples depends on both the means and the standard deviations. Very different means can occur by chance if there is great variation among the individual samples. In order to account for the variation, we take the difference of the sample means, X ¯ 1 X ¯ 1 X ¯ 2 X ¯ 2 , and divide by the standard error in order to standardize the difference. The result is a t-score test statistic.

Because we do not know the population standard deviations, we estimate them using the two sample standard deviations from our independent samples. For the hypothesis test, we calculate the estimated standard deviation, or standard error, of the difference in sample means, X ¯ 1 X ¯ 1 X ¯ 2 X ¯ 2 .

The standard error is: ( s 1 ) 2 n 1 + ( s 2 ) 2 n 2 ( s 1 ) 2 n 1 + ( s 2 ) 2 n 2

The test statistic (t-score) is calculated as follows:

( x ¯ 1 x ¯ 2 )( μ 1 μ 2 ) ( s 1 ) 2 n 1 + ( s 2 ) 2 n 2 ( x ¯ 1 x ¯ 2 )( μ 1 μ 2 ) ( s 1 ) 2 n 1 + ( s 2 ) 2 n 2
where:
  • s1 and s2, the sample standard deviations, are estimates of σ1 and σ2, respectively.
  • σ1 and σ1 are the unknown population standard deviations.
  • x ¯ 1 x ¯ 1 and x ¯ 2 x ¯ 2 are the sample means. μ1 and μ2 are the population means.

The number of degrees of freedom (df) requires a somewhat complicated calculation. However, a computer or calculator calculates it easily. The df are not always a whole number. The test statistic calculated previously is approximated by the Student's t-distribution with df as follows:

Degrees of freedom df= ( ( s 1 ) 2 n 1 + ( s 2 ) 2 n 2 ) 2 ( 1 n 1 1 ) ( ( s 1 ) 2 n 1 ) 2 +( 1 n 2 1 ) ( ( s 2 ) 2 n 2 ) 2 df= ( ( s 1 ) 2 n 1 + ( s 2 ) 2 n 2 ) 2 ( 1 n 1 1 ) ( ( s 1 ) 2 n 1 ) 2 +( 1 n 2 1 ) ( ( s 2 ) 2 n 2 ) 2

When both sample sizes n1 and n2 are five or larger, the Student's t approximation is very good. Notice that the sample variances (s1)2 and (s2)2 are not pooled. (If the question comes up, do not pool the variances.)

NOTE

It is not necessary to compute this by hand. A calculator or computer easily computes it.

Example 10.1 Independent groups

The average amount of time boys and girls aged seven to 11 spend playing sports each day is believed to be the same. A study is done and data are collected, resulting in the data in Table 10.1. Each populations has a normal distribution.

Sample Size Average Number of Hours Playing Sports Per Day Sample Standard Deviation
Girls 9 2 0.8660.866
Boys 16 3.2 1.00
Table 10.1

Is there a difference in the mean amount of time boys and girls aged seven to 11 play sports each day? Test at the 5% level of significance.

Solution 10.1

The population standard deviations are not known. Let g be the subscript for girls and b be the subscript for boys. Then, μg is the population mean for girls and μb is the population mean for boys. This is a test of two independent groups, two population means.

Random variable: X ¯ g X ¯ b X ¯ g X ¯ b = difference in the sample mean amount of time girls and boys play sports each day.
H0: μg = μb  H0: μgμb = 0
Ha: μgμb  Ha: μgμb ≠ 0
The words "the same" tell you H0 has an "=". Since there are no other words to indicate Ha, assume it says "is different." This is a two-tailed test.

Distribution for the test: Use tdf where df is calculated using the df formula for independent groups, two population means. Using a calculator, df is approximately 18.8462. Do not pool the variances.

Calculate the p-value using a Student's t-distribution: p-value = 0.0054

Graph:

This is a normal distribution curve representing the difference in the average amount of time girls and boys play sports all day. The mean is equal to zero, and the values -1.2, 0, and 1.2 are labeled on the horizontal axis. Two vertical lines extend from -1.2 and 1.2 to the curve. The region to the left of x = -1.2 and the region to the right of x = 1.2 are shaded to represent the p-value. The area of each region is 0.0028.
Figure 10.2


s g = 0.866 s g = 0.866
s b = 1 s b =1
So, x ¯ g x ¯ b x ¯ g x ¯ b = 2 – 3.2 = –1.2
Half the p-value is below –1.2 and half is above 1.2.

Make a decision: Since α > p-value, reject H0. This means you reject μg = μb. The means are different.

Using the TI-83, 83+, 84, 84+ Calculator

Press STAT. Arrow over to TESTS and press 4:2-SampTTest. Arrow over to Stats and press ENTER. Arrow down and enter 2 for the first sample mean, 0.866 0.866 for Sx1, 9 for n1, 3.2 for the second sample mean, 1 for Sx2, and 16 for n2. Arrow down to μ1: and arrow to does not equal μ2. Press ENTER. Arrow down to Pooled: and No. Press ENTER. Arrow down to Calculate and press ENTER. The p-value is p = 0.0054, the dfs are approximately 18.8462, and the test statistic is -3.14. Do the procedure again but instead of Calculate do Draw.

Conclusion: At the 5% level of significance, the sample data show there is sufficient evidence to conclude that the mean number of hours that girls and boys aged seven to 11 play sports per day is different (mean number of hours boys aged seven to 11 play sports per day is greater than the mean number of hours played by girls OR the mean number of hours girls aged seven to 11 play sports per day is greater than the mean number of hours played by boys).

Try It 10.1

Two samples are shown in Table 10.2. Both have normal distributions. The means for the two populations are thought to be the same. Is there a difference in the means? Test at the 5% level of significance.

Sample Size Sample Mean Sample Standard Deviation
Population A 25 5 1
Population B 16 4.7 1.2
Table 10.2

NOTE

When the sum of the sample sizes is larger than 30 (n1 + n2 > 30) you can use the normal distribution to approximate the Student's t.

Example 10.2

A study is done by a community group in two neighboring colleges to determine which one graduates students with more math classes. College A samples 11 graduates. Their average is four math classes with a standard deviation of 1.5 math classes. College B samples nine graduates. Their average is 3.5 math classes with a standard deviation of one math class. The community group believes that a student who graduates from college A has taken more math classes, on the average. Both populations have a normal distribution. Test at a 1% significance level. Answer the following questions.

a. Is this a test of two means or two proportions?

Solution 10.2

a. two means

b. Are the populations standard deviations known or unknown?

Solution 10.2

b. unknown

c. Which distribution do you use to perform the test?

Solution 10.2

c. Student's t

d. What is the random variable?

Solution 10.2

d. X ¯ A - X ¯ B X ¯ A - X ¯ B

e. What are the null and alternate hypotheses? Write the null and alternate hypotheses in words and in symbols.

Solution 10.2

e.

  • H o : μ A μ B H o : μ A μ B
  • H a : μ A > μ B H a : μ A > μ B



f. Is this test right-, left-, or two-tailed?

Solution 10.2

f.

This is a normal distribution curve with mean equal to 0. A vertical line near the tail of the curve to the right of zero extends from the axis to the curve. The region under the curve to the right of the line is shaded.
Figure 10.3


right

g. What is the p-value?

Solution 10.2

g. 0.1928

h. Do you reject or not reject the null hypothesis?

Solution 10.2

h. Do not reject.

i. Conclusion:

Solution 10.2

i. At the 1% level of significance, from the sample data, there is not sufficient evidence to conclude that a student who graduates from college A has taken more math classes, on the average, than a student who graduates from college B.

Try It 10.2

A study is done to determine if Company A retains its workers longer than Company B. Company A samples 15 workers, and their average time with the company is five years with a standard deviation of 1.2. Company B samples 20 workers, and their average time with the company is 4.5 years with a standard deviation of 0.8. The populations are normally distributed.

  1. Are the population standard deviations known?
  2. Conduct an appropriate hypothesis test. At the 5% significance level, what is your conclusion?

Example 10.3

A professor at a large community college wanted to determine whether there is a difference in the means of final exam scores between students who took his statistics course online and the students who took his face-to-face statistics class. He believed that the mean of the final exam scores for the online class would be lower than that of the face-to-face class. Was the professor correct? The randomly selected 30 final exam scores from each group are listed in Table 10.3 and Table 10.4.

67.6 41.2 85.3 55.9 82.4 91.2 73.5 94.1 64.7 64.7
70.6 38.2 61.8 88.2 70.6 58.8 91.2 73.5 82.4 35.5
94.1 88.2 64.7 55.9 88.2 97.1 85.3 61.8 79.4 79.4
Table 10.3 Online Class
77.9 95.3 81.2 74.1 98.8 88.2 85.9 92.9 87.1 88.2
69.4 57.6 69.4 67.1 97.6 85.9 88.2 91.8 78.8 71.8
98.8 61.2 92.9 90.6 97.6 100 95.3 83.5 92.9 89.4
Table 10.4 Face-to-face Class

Is the mean of the Final Exam scores of the online class lower than the mean of the Final Exam scores of the face-to-face class? Test at a 5% significance level. Answer the following questions:

  1. Is this a test of two means or two proportions?
  2. Are the population standard deviations known or unknown?
  3. Which distribution do you use to perform the test?
  4. What is the random variable?
  5. What are the null and alternative hypotheses? Write the null and alternative hypotheses in words and in symbols.
  6. Is this test right, left, or two tailed?
  7. What is the p-value?
  8. Do you reject or not reject the null hypothesis?
  9. At the ___ level of significance, from the sample data, there ______ (is/is not) sufficient evidence to conclude that ______.

(See the conclusion in Example 10.2, and write yours in a similar fashion)

Using the TI-83, 83+, 84, 84+ Calculator

First put the data for each group into two lists (such as L1 and L2). Press STAT. Arrow over to TESTS and press 4:2SampTTest. Make sure Data is highlighted and press ENTER. Arrow down and enter L1 for the first list and L2 for the second list. Arrow down to μ1: and arrow to ≠ μ2 (does not equal). Press ENTER. Arrow down to Pooled: No. Press ENTER. Arrow down to Calculate and press ENTER.

Note

Be careful not to mix up the information for Group 1 and Group 2!

Solution 10.3
  1. two means
  2. unknown
  3. Student's t
  4. X ¯ 1 X ¯ 2 X ¯ 1 X ¯ 2
    1. H0: μ1 = μ2 Null hypothesis: the means of the final exam scores are equal for the online and face-to-face statistics classes.
    2. Ha: μ1 < μ2 Alternative hypothesis: the mean of the final exam scores of the online class is less than the mean of the final exam scores of the face-to-face class.
  5. left-tailed
  6. p-value = 0.0011
    This is a normal distribution curve with mean equal to zero. A vertical line near the tail of the curve to the left of zero extends from the axis to the curve. The region under the curve to the left of the line is shaded representing p-value = 0.0011.
    Figure 10.4
  7. Reject the null hypothesis
  8. The professor was correct. The evidence shows that the mean of the final exam scores for the online class is lower than that of the face-to-face class.
    At the 5% level of significance, from the sample data, there is (is/is not) sufficient evidence to conclude that the mean of the final exam scores for the online class is less than the mean of final exam scores of the face-to-face class.

Cohen's Standards for Small, Medium, and Large Effect SizesCohen's d is a measure of effect size based on the differences between two means. Cohen’s d, named for United States statistician Jacob Cohen, measures the relative strength of the differences between the means of two populations based on sample data. The calculated value of effect size is then compared to Cohen’s standards of small, medium, and large effect sizes.

Size of effect d
Small 0.2
medium 0.5
Large 0.8
Table 10.5 Cohen's Standard Effect Sizes

Cohen's d is the measure of the difference between two means divided by the pooled standard deviation: d= x ¯ 1 x ¯ 2 s pooled d= x ¯ 1 x ¯ 2 s pooled where s pooled = ( n 1 1) s 1 2 +( n 2 1) s 2 2 n 1 + n 2 2 s pooled = ( n 1 1) s 1 2 +( n 2 1) s 2 2 n 1 + n 2 2

Example 10.4

Calculate Cohen’s d for Example 10.2. Is the size of the effect small, medium, or large? Explain what the size of the effect means for this problem.

Solution 10.4

μ1 = 4 s1 = 1.5 n1 = 11
μ2 = 3.5 s2 = 1 n2 = 9
d = 0.384
The effect is small because 0.384 is between Cohen’s value of 0.2 for small effect size and 0.5 for medium effect size. The size of the differences of the means for the two colleges is small indicating that there is not a significant difference between them.

Example 10.5

Calculate Cohen’s d for Example 10.3. Is the size of the effect small, medium or large? Explain what the size of the effect means for this problem.

Solution 10.5

d = 0.834; Large, because 0.834 is greater than Cohen’s 0.8 for a large effect size. The size of the differences between the means of the Final Exam scores of online students and students in a face-to-face class is large indicating a significant difference.

Try It 10.5

Weighted alpha is a measure of risk-adjusted performance of stocks over a period of a year. A high positive weighted alpha signifies a stock whose price has risen while a small positive weighted alpha indicates an unchanged stock price during the time period. Weighted alpha is used to identify companies with strong upward or downward trends. The weighted alpha for the top 30 stocks of banks in the northeast and in the west as identified by Nasdaq on May 24, 2013 are listed in Table 10.6 and Table 10.7, respectively.

94.2 75.2 69.6 52.0 48.0 41.9 36.4 33.4 31.5 27.6
77.3 71.9 67.5 50.6 46.2 38.4 35.2 33.0 28.7 26.5
76.3 71.7 56.3 48.7 43.2 37.6 33.7 31.8 28.5 26.0
Table 10.6 Northeast
126.0 70.6 65.2 51.4 45.5 37.0 33.0 29.6 23.7 22.6
116.1 70.6 58.2 51.2 43.2 36.0 31.4 28.7 23.5 21.6
78.2 68.2 55.6 50.3 39.0 34.1 31.0 25.3 23.4 21.5
Table 10.7 West

Is there a difference in the weighted alpha of the top 30 stocks of banks in the northeast and in the west? Test at a 5% significance level. Answer the following questions:

  1. Is this a test of two means or two proportions?
  2. Are the population standard deviations known or unknown?
  3. Which distribution do you use to perform the test?
  4. What is the random variable?
  5. What are the null and alternative hypotheses? Write the null and alternative hypotheses in words and in symbols.
  6. Is this test right, left, or two tailed?
  7. What is the p-value?
  8. Do you reject or not reject the null hypothesis?
  9. At the ___ level of significance, from the sample data, there ______ (is/is not) sufficient evidence to conclude that ______.
  10. Calculate Cohen’s d and interpret it.
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