Homework 1: One-sample Hypothesis Testing with known population variance
- Test the claim that the true mean diameter of a manufactured bolt is 30mm. Assume population standard deviation equals 0.8mm. Tolerate type I error =0.05. A sample of 100 bolts is collected. The sample mean is 29.84. Please use both the P-value method and the critical value method to test your understanding of the two methods.
Answer to question 1 with critical value method
- The value of interest is
30mm - State the appropriate null and alternative hypotheses
H0: µ=30 H1: µ≠30 This is a two tail test - Specify the desired level of significance
Suppose that = 0.05 - (only when using critical value method) Determine the critical values
For = 0.05 the critical Z values are ±1.96 (two tails, two values)
The null use the equal sign, so the level of significance is divided by 2, and assigned equally to the left and the right side. - Determine the appropriate test statistic
σ is assumed known so this is a Z test.
Compute the test statistic according to the sample information - Last step, make your decision
(only when using critical value method)
ZSTAT = -2.0 < -1.96, so the test statistic is in the rejection region
Answer to question 1 with p-value method
- The value of interest is
30mm - State the appropriate null and alternative hypotheses
H0: µ=30 H1: µ≠30 This is a two tail test - Specify the desired level of significance
Suppose that = 0.05 - Determine the appropriate test statistic
σ is assumed known so this is a Z test.
Compute the test statistic according to the sample information - (only when using p-value method) Determine the p-value for ONE of the two sides. Then multiply by 2.
P-value = 0.0228 *2 - Last step, make your decision
(when using p-value method)
p-value=0.0456<0.05=, so we reject - Test the claim that the true mean diameter of a manufactured bolt is larger than 30mm. Assume population standard deviation equals 0.8mm. Tolerate type I error =0.05. A sample of 100 bolts is collected. The sample mean is 29.84. Please use both the P-value method and the critical value method to test your understanding of the two methods.
Answer to question 1 with critical value method
- The value of interest is
30mm - State the appropriate null and alternative hypotheses
H0: µ<=30 H1: µ>30
This is a ONE tail test. We also flipped the null and alternative hypothesis to keep the equal sign in the null. - Specify the desired level of significance
Suppose that = 0.05 - (only when using critical value method) Determine the critical values
For = 0.05 the critical Z values are 1.64
(One-sided one value! Upper tail test, so positive value)
Determine the appropriate test statistic
σ is assumed known so this is a Z test. - Compute the test statistic according to the sample information
- Last step, make your decision
(only when using critical value method)
ZSTAT = -2.0 < 1.64, so the test statistic is in the do not reject region. The population mean is smaller than or equal to 30.
Answer to question 1 with p-value method
- The value of interest is
30mm - State the appropriate null and alternative hypotheses
H0: µ<=30 H1: µ>30 - Specify the desired level of significance
Suppose that = 0.05 - Determine the appropriate test statistic
σ is assumed known so this is a Z test.
Compute the test statistic according to the sample information - (only when using p-value method) Determine the p-value for ONE SIDE!
P-value =Prob(Z>=-2)=0.9772 - Last step, make your decision
(when using p-value method)
p-value=0.9772>0.05= , so we do not reject