Two Population Means with Known Standard Deviations Quiz
The question sheet
Reveal any answer as you study-
In the floor wax test, what does the random variable X̄₁ – X̄₂ represent?
- Ratio of the standard deviations
- Difference in sample sizes
- Difference in mean months waxes last
- Sum of the two population means
Reveal answer
Answer: Difference in mean months waxes last
Source evidence
PDF page 585: Solution 10.6 This is a test of two independent groups, two population means, population standard deviations known. ̄ ̄ Random Variable: X – X = difference in the mean number of months the competing floor waxes last. 1 2
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In the floor wax example, what was the calculated value of X̄₁ – X̄₂?
- 0.1799
- 2.9
- 0.1
- 0.9157
Reveal answer
Answer: 0.1
Source evidence
PDF page 586: X – X = 3 – 2.9 = 0.1
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For the floor wax test, what p-value was found using the normal distribution?
- 0.1799
- 0.05
- 0.9157
- 0.4955
Reveal answer
Answer: 0.1799
Source evidence
PDF page 585: Calculate the p-value using the normal distribution: p-value = 0.1799 Graph:
PDF page 586: 1 2 Compare α and the p-value: α = 0.05 and p-value = 0.1799. Therefore, α < p-value. Make a decision: Since α < p-value, do not reject H0. Conclusion: At the 5% level of significance, from the sample data, there is not sufficient evidence to conclude that the mean time wax 1 lasts is longer (wax 1 is more effective) than the mean time wax 2 lasts.
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In the floor wax test, comparing α = 0.05 and p-value = 0.1799, the decision is to:
- Test again at 1%
- Do not reject H0
- Fail to compute
- Reject H0
Reveal answer
Answer: Do not reject H0
Source evidence
PDF page 586: 1 2 Compare α and the p-value: α = 0.05 and p-value = 0.1799. Therefore, α < p-value. Make a decision: Since α < p-value, do not reject H0. Conclusion: At the 5% level of significance, from the sample data, there is not sufficient evidence to conclude that the mean time wax 1 lasts is longer (wax 1 is more effective) than the mean time wax 2 lasts.
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For independent means with known population standard deviations, both populations must be:
- Skewed
- Uniform
- Exponential
- Normal
Reveal answer
Answer: Normal
Source evidence
PDF page 584: Even though this situation is not likely (knowing the population standard deviations is not likely), the following example illustrates hypothesis testing for independent means, known population standard deviations. The sampling distribution for the difference between the means is normal and both populations must be normal. The random variable is X – X . The 1 2 normal distribution has the following format: Normal distribution is: ⎡
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In the floor wax test, what test statistic (z-score) was found?
- 0.9157
- 0.1
- 0.1799
- 0.05
Reveal answer
Answer: 0.9157
Source evidence
PDF page 586: Press STAT. Arrow over to TESTS and press 3:2-SampZTest. Arrow over to Stats and press ENTER. Arrow down and enter .33 for sigma1, .36 for sigma2, 3 for the first sample mean, 20 for n1, 2.9 for the second sample mean, and 20 for n2. Arrow down to μ1: and arrow to > μ2. Press ENTER. Arrow down to Calculate and press ENTER. The p-value is p = 0.1799 and the test statistic is 0.9157. Do the procedure again, but instead of Calculate do Draw.
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On the TI, which command performs the floor wax two-means test?
- 1-PropZTest
- 2-SampZTest
- ANOVA
- 2-SampTTest
Reveal answer
Answer: 2-SampZTest
Source evidence
PDF page 586: Press STAT. Arrow over to TESTS and press 3:2-SampZTest. Arrow over to Stats and press ENTER. Arrow down and enter .33 for sigma1, .36 for sigma2, 3 for the first sample mean, 20 for n1, 2.9 for the second sample mean, and 20 for n2. Arrow down to μ1: and arrow to > μ2. Press ENTER. Arrow down to Calculate and press ENTER. The p-value is p = 0.1799 and the test statistic is 0.9157. Do the procedure again, but instead of Calculate do Draw.
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The phrase 'wax 1 is more effective than wax 2' makes the floor wax test a:
- Non-directional test
- Left-tailed test
- Right-tailed test
- Two-tailed test
Reveal answer
Answer: Right-tailed test
Source evidence
PDF page 585: The words "is more effective" says that wax 1 lasts longer than wax 2, on average. "Longer" is a “>” symbol and goes into Ha. Therefore, this is a right-tailed test. Distribution for the test: The population standard deviations are known so the distribution is normal. Using the formula, the distribution is: 2
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In the floor wax test, why is the mean of the normal distribution zero?
- Sample sizes are equal
- Both means are equal always
- Standard deviations are known
- μ1 ≤ μ2 so μ1 – μ2 ≤ 0
Reveal answer
Answer: μ1 ≤ μ2 so μ1 – μ2 ≤ 0
Source evidence
PDF page 585: Since μ1 ≤ μ2 then μ1 – μ2 ≤ 0 and the mean for the normal distribution is zero.
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How were the twenty floors assigned to test each floor wax?
- By age of floor
- Randomly
- By floor location
- By wax cost
Reveal answer
Answer: Randomly
Source evidence
PDF page 585: Independent groups, population standard deviations known: The mean lasting time of two competing floor waxes is to be compared. Twenty floors are randomly assigned to test each wax. Both populations have a normal distributions. The data are recorded in Table 10.8.
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In the senator age example, how many Republican senators were randomly selected?
- 10
- 20
- 61
- 30
Reveal answer
Answer: 30
Source evidence
PDF page 587: average. On May 26 2013, the mean age of 30 randomly selected Republican Senators was 61 years 247 days old (61.675 years) with a standard deviation of 10.17 years. The mean age of 30 randomly selected Democratic senators was 61 years 257 days old (61.704 years) with a standard deviation of 9.55 years. Do the data indicate that Democratic senators are older than Republican senators, on average? Test at a 5% level of significance.
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In the senator study, the standard deviation of Democratic senators' ages was:
- 61.7 years
- 9.55 years
- 10.17 years
- 0.36 years
Reveal answer
Answer: 9.55 years
Source evidence
PDF page 587: average. On May 26 2013, the mean age of 30 randomly selected Republican Senators was 61 years 247 days old (61.675 years) with a standard deviation of 10.17 years. The mean age of 30 randomly selected Democratic senators was 61 years 257 days old (61.704 years) with a standard deviation of 9.55 years. Do the data indicate that Democratic senators are older than Republican senators, on average? Test at a 5% level of significance.
PDF page 587: ̄ ̄ (9.55) (10.17) + ] groups. Using the formula, the distribution is: X – X ∼ N[0, 1 2
Introductory Statistics
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