Statistics questions
MTH 245 Lesson 20 Notes One-Sample Test for the Mean
The methods in this section presume that the data fit at least one of the following two criteria:
1. 𝑛𝑛 ≥ 30, or 2. The original random variable is normally distributed.
If 𝑛𝑛 < 30 and the original random variable is not normally distributed, we need to use bootstrapping or some other non-parametric method. (Bootstrapping will be covered in Section 8-3.) When the Central Limit Theorem Applies and 𝜎𝜎 is Known In the rare case where we know the value of 𝜎𝜎 (or can make an assumption of its value with reasonable certainty), we can make use of the Central Limit Theorem and calculate a p-value using the normal distribution. When the Central Limit Theorem Applies and 𝜎𝜎 is Unknown Most of the time, we won't know the value of 𝜎𝜎. In this case, we need to substitute the sample standard deviation 𝑠𝑠 for 𝜎𝜎. Unfortunately, 𝑠𝑠 is a biased estimator of 𝜎𝜎, so if we use it with the normal distribution, we'll wind up with a p-value that is too large. We compensate for this by using the 𝑡𝑡 distribution (aka Student's t) instead of the normal distribution. The 𝑡𝑡 distribution has greater variability than the standard normal distribution, and therefore compensates for the tendency for 𝑠𝑠 to underestimate 𝜎𝜎.
Comparison of the standard normal and 𝑡𝑡 distributions (3 degrees of freedom)
The following table summarizes the foregoing discussion:
When… and… use… and the StatCrunch
menu…
𝑥𝑥 is normal OR 𝑛𝑛 ≥ 30
𝜎𝜎 is unknown t Distribution Stat T-Stats
𝜎𝜎 is known Normal
Distribution Stat Z-Stats
𝑥𝑥 is not normal AND 𝑛𝑛 < 30
Bootstrapping/non-parametric methods
N/A
One-Sample Test for the Mean Using StatCrunch To use StatCrunch to conduct a one-sample hypothesis test for 𝜇𝜇 when sample statistics (�̅�𝑥, 𝑠𝑠, and 𝑛𝑛) are already available:
1. Open a blank data table. 2. For the 𝑡𝑡 distribution, click Stat T Stats One Sample With
Summary. For the normal distribution, click Stat Z Stats One Sample With Summary.
3. Fill in the sample statistics. 4. Under "Perform:", leave the radio button
at "Hypothesis test for 𝜇𝜇" (the default). 5. Fill in the null hypothesis value and the
alternative hypothesis operator. 6. Click "Compute!".
To use StatCrunch to conduct a one-sample hypothesis test for 𝜇𝜇 using raw data:
1. Import/enter the data. 2. For the 𝑡𝑡 distribution, click Stat T Stats One Sample With
Data. For the normal distribution, click Stat Z Stats One Sample With Data.
3. Select the appropriate data column. 4. Under "Perform:", leave the radio button at
"Hypothesis test for 𝜇𝜇" (the default). 5. Fill in the null hypothesis value and the
alternative hypothesis operator. 6. Click "Compute!".
Example 1 (𝜎𝜎 unknown): The Department of Motor Vehicles office in a certain city claims that its mean wait time for service is at most 14 minutes. A random sample of 70 customers had a mean wait time of 13 minutes with a standard deviation of 3.5 minutes. At significance level 𝛼𝛼 = 0.10, is there enough evidence to reject the office's claim?
Identify the correct null and alternative hypotheses.
𝐻𝐻0: 𝜇𝜇 = 14 min (Original claim: 𝜇𝜇 ≤ 14 min) 𝐻𝐻𝐴𝐴: 𝜇𝜇 > 14 min
What is the p-value? (Round to three decimal places as needed.) 0.990
State and interpret the appropriate decision for this hypothesis test.
Since the p-value = 0.990 > 𝛼𝛼 = 0.10, we fail to reject 𝐻𝐻0. There is insufficient evidence to reject the DMV's claim that its mean wait time for service is at most 14 minutes.
Example 2 (𝜎𝜎 unknown): A research team believes that the mean red blood cell count for adult males (in millions of cells per microliter) is 4.950. The RBC Counts data set contains red blood cell counts for 40 men who took part in the team's clinical study. Use this data set to test the team's claim at significance level 𝛼𝛼 = 0.05.
Identify the correct null and alternative hypotheses.
𝐻𝐻0: 𝜇𝜇 = 4.950 106 cells/dL (Original claim: 𝜇𝜇 = 4.950 106 cells/dL) 𝐻𝐻𝐴𝐴: 𝜇𝜇 ≠ 4.950 106 cells/dL
What is the p-value? (Round to three decimal places as needed.) 0.058
State and interpret the appropriate decision for this hypothesis test.
Since the p-value = 0.058 > 𝛼𝛼 = 0.05, we fail to reject 𝐻𝐻0. There is insufficient evidence to reject the research team's claim that the mean red blood cell count for adult males = 4.950 106 cells/dL.
Example 3 (𝜎𝜎 known): A certain drink container has a labeled weight of 12 fluid ounces. The manufacturer claims that the weights of the individual containers have mean 𝜇𝜇 = 12.00 fl oz and standard deviation 𝜎𝜎 = 0.11 fl oz. Suppose a quality control sample of 36 randomly selected containers has a mean weight of 12.19 fl oz. At significance level 𝛼𝛼 = 0.01, is there enough evidence to reject the manufacturer's claim?
Identify the correct null and alternative hypotheses.
𝐻𝐻0: 𝜇𝜇 = 12.00 fl oz (Original claim: 𝜇𝜇 = 12.00 fl oz) 𝐻𝐻𝐴𝐴: 𝜇𝜇 ≠ 12.00 fl oz
What is the p-value? (Round to three decimal places as needed.) 0.000
State and interpret the appropriate decision for this hypothesis test.
Since the p-value = 0.000 < 𝛼𝛼 = 0.01, we reject 𝐻𝐻0. There is sufficient evidence to reject the manufacturer's claim that the weights of the individual containers have mean 𝜇𝜇 = 12.00 fl oz.
Example 4 (𝜎𝜎 unknown): An environmental activist group claims that the mean pH level of the water in a certain municipal reservoir is less than 6.5, the minimum pH that meets EPA safety standards. In the resulting water quality analysis, 39 randomly selected water samples are found to have a mean pH level of 6.7 with a standard deviation of 0.35. At significance level 𝛼𝛼 = 0.05, is there enough evidence to support the group's claim?
Identify the correct null and alternative hypotheses.
𝐻𝐻0: 𝜇𝜇 = 6.5 𝐻𝐻𝐴𝐴: 𝜇𝜇 < 6.5 (Original claim: 𝜇𝜇 < 6.5)
What is the p-value? (Round to three decimal places as needed.) 1.000
State and interpret the appropriate decision for this hypothesis test.
Since the p-value = 1.000 > 𝛼𝛼 = 0.05, we fail to reject 𝐻𝐻0. There is insufficient evidence to support the environmental group's claim that the mean pH level of the water in a certain municipal reservoir is less than 6.5.