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1. (a) A statistician randomly sampled 100 observations and found Xˉ=106\bar{X}=106 ...
Apr 21, 2024
1. (a) A statistician randomly sampled 100 observations and found Xˉ=106\bar{X}=106 and s=35s=35. Calculate the tt statistic and pp value for testing. H_{0}: \mu=100 \text { against } H_{A}: \mu>100 \text {. } Carry out the test at the 1%1 \% level of significance. (b) Repeat part (a), with s=25s=25. (c) Repeat part (a), with s=15s=15. 2. Repeat question 1 using HA:μ≠100H_{A}: \mu \neq 100.
Solution by Steps
step 1
Calculate the tt statistic using the formula t=Xˉ−μ0s/nt = \frac{\bar{X} - \mu_0}{s/\sqrt{n}} where Xˉ=106\bar{X} = 106, μ0=100\mu_0 = 100, s=35s = 35, and n=100n = 100
step 2
Substitute the values into the formula: t=106−10035/100=635/10=63.5=1.7143t = \frac{106 - 100}{35/\sqrt{100}} = \frac{6}{35/10} = \frac{6}{3.5} = 1.7143
step 3
Determine the pp value using a tt-distribution table or software for a one-tailed test with 99 degrees of freedom
step 4
Compare the pp value to the level of significance, 0.010.01. If p < 0.01, reject H0H_0
Answer
The tt statistic is approximately 1.71431.7143. The pp value must be looked up or calculated using statistical software. If p < 0.01, reject H0H_0.
Key Concept
Calculating the tt statistic for a one-sample tt test
Explanation
The tt statistic measures how many standard errors the sample mean is from the null hypothesis mean. It is used to determine the pp value for the test.
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step 1
Calculate the tt statistic with s=25s = 25 using the same formula: t=Xˉ−μ0s/nt = \frac{\bar{X} - \mu_0}{s/\sqrt{n}}
step 2
Substitute the values: t=106−10025/100=625/10=62.5=2.4t = \frac{106 - 100}{25/\sqrt{100}} = \frac{6}{25/10} = \frac{6}{2.5} = 2.4
step 3
Determine the pp value using a tt-distribution table or software for a one-tailed test with 99 degrees of freedom
step 4
Compare the pp value to the level of significance, 0.010.01. If p < 0.01, reject H0H_0
Answer
The tt statistic is 2.42.4. The pp value must be looked up or calculated using statistical software. If p < 0.01, reject H0H_0.
Key Concept
Calculating the tt statistic for a one-sample tt test with different standard deviations
Explanation
A smaller standard deviation increases the tt statistic, making it more likely to reject the null hypothesis if the mean difference is significant.
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step 1
Calculate the tt statistic with s=15s = 15 using the same formula: t=Xˉ−μ0s/nt = \frac{\bar{X} - \mu_0}{s/\sqrt{n}}
step 2
Substitute the values: t=106−10015/100=615/10=61.5=4t = \frac{106 - 100}{15/\sqrt{100}} = \frac{6}{15/10} = \frac{6}{1.5} = 4
step 3
Determine the pp value using a tt-distribution table or software for a one-tailed test with 99 degrees of freedom
step 4
Compare the pp value to the level of significance, 0.010.01. If p < 0.01, reject H0H_0
Answer
The tt statistic is 44. The pp value must be looked up or calculated using statistical software. If p < 0.01, reject H0H_0.
Key Concept
Impact of standard deviation on the tt statistic
Explanation
A decrease in standard deviation results in a higher tt statistic, which can lead to a stronger rejection of the null hypothesis if the mean difference is significant.
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step 1
For the two-sided tt test with s=35s = 35, calculate the tt statistic using the same formula
step 2
The tt statistic remains the same as in the one-tailed test: 1.71431.7143
step 3
Determine the pp value for a two-tailed test using a tt-distribution table or software with 99 degrees of freedom
step 4
The pp value for a two-tailed test is double that of a one-tailed test. Compare 2p2p to the level of significance, 0.010.01. If 2p < 0.01, reject H0H_0
Answer
The tt statistic is 1.71431.7143. The pp value for a two-tailed test must be looked up or calculated using statistical software. If 2p < 0.01, reject H0H_0.
Key Concept
Calculating the tt statistic for a two-sided tt test
Explanation
In a two-sided test, the pp value is doubled because the alternative hypothesis considers both directions of the mean difference.
---
step 1
For the two-sided tt test with s=25s = 25, calculate the tt statistic using the same formula
step 2
The tt statistic remains the same as in the one-tailed test: 2.42.4
step 3
Determine the pp value for a two-tailed test using a tt-distribution table or software with 99 degrees of freedom
step 4
The pp value for a two-tailed test is double that of a one-tailed test. Compare 2p2p to the level of significance, 0.010.01. If 2p < 0.01, reject H0H_0
Answer
The tt statistic is 2.42.4. The pp value for a two-tailed test must be looked up or calculated using statistical software. If 2p < 0.01, reject H0H_0.
Key Concept
Calculating the tt statistic for a two-sided tt test with different standard deviations
Explanation
The pp value for a two-tailed test accounts for the possibility of the sample mean being significantly higher or lower than the null hypothesis mean.
---
step 1
For the two-sided tt test with s=15s = 15, calculate the tt statistic using the same formula
step 2
The tt statistic remains the same as in the one-tailed test: 44
step 3
Determine the pp value for a two-tailed test using a tt-distribution table or software with 99 degrees of freedom
step 4
The pp value for a two-tailed test is double that of a one-tailed test. Compare 2p2p to the level of significance, 0.010.01. If 2p < 0.01, reject H0H_0
Answer
The tt statistic is 44. The pp value for a two-tailed test must be looked up or calculated using statistical software. If 2p < 0.01, reject H0H_0.
Key Concept
Impact of standard deviation on the tt statistic in a two-sided tt test
Explanation
A smaller standard deviation in a two-sided test increases the tt statistic, which affects the pp value and the decision to reject or fail to reject the null hypothesis.
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