AP StatisticsName: ______
NotesDate: ______
Lesson 11.3A: A Test of Hypothesis for a Population Mean
Big IdeaHow can we assess the truth of a claim about a population parameter using
information gained in a simple random sample from that population?
Objectives: R, S, T, U
I.One-Sided t-Test of Hypothesis of
Example #1:Are young women delaying marriage and marrying at a later age?
This question was addressed in a report issued by the Census Bureau (Associated Press, June, 2001). The report stated that in 1990 (based on census results) the mean age of brides marrying for the first time was 23.9 years. In 2001, based on a random sample of 100 young women, the mean was 25.8 years and the standard deviation was 6.4 years. Is there sufficient evidence to support the claim that women are now marrying later in life? Test using a significance level of = .05.
Example #2: A growing concern of employers is time spent in activities like surfing the Internet and emailing friends during work hours. The San Luis Obispo Tribune summarized the findings from a survey of a large sample of workers in an article that ran under the headline “Who Goofs Off 2 Hours a Day? Most Workers, Survey Says” (August 3, 2006). Suppose that the CEO of a large company wants to determine whether the average amount of wasted time during an eight-hour work day for employees of her company is less than the reported 120 minutes. Each person in a random sample of 10 employees was contacted and asked about daily wasted time at work. (Participants would probably have to be guaranteed anonymity to obtain truthful responses! What type of bias may be an issue here?) The resulting data follow:
108112117130111131113113105128
Do these data provide sufficient evidence that the mean wasted time for this company is less than 120 minutes? Use a significance level of 0.05.
II.Two-Sided t-Test of Hypothesis of
Example #3: During an angiogram, heart problems can be examined via a small tube
(a catheter) threaded into the heart from a vein in the patient’s leg. It’s important that the company that manufactures the catheter maintain a diameter of 2.00 mm. On a given day, quality control personnel randomly sample 50 catheters and record the sample mean diameter to be 1.995 mm and the sample standard deviation to be 0.016 mm. Do they have sufficient evidence to conclude that the mean diameter of all catheters produced that day differ from 2.00 mm?
a. Are the data statistically significant at the 0.05 level?
b. Are the data statistically significant at the 0.01 level?
c. Construct a 95% confidence interval for the mean diameter of all catheters produced
on the day in question. Is this confidence interval consistent with the decision you
reached when performing a two-sided test at the 0.05 level of significance?
d. Construct a 99% confidence interval for the mean diameter of all catheters produced
on the day in question. Is the confidence interval consistent with the decision you
reached when performing a two-sided test at the 0.01 level of significance?
Recall: Many statisticians prefer to use confidence intervals in studies because they
provide an estimate of the parameter of interest and not just a “reject or fail to reject “ conclusion. The idea is pretty straightforward:
The duality between two-sided significance tests and confidence intervals:If the parameter value given in the null hypothesis falls inside the confidence interval, then that value is plausible. Fail to Reject .
If the parameter value lands outside the confidence interval, then we have good reason to doubt. Reject .
Example #4:Many consumers pay careful attention to stated nutritional contents on packaged foods when making purchases. It is therefore important that the information on packages be accurate. A random sample of n =12 frozen dinners of a certain type was selected from production during a particular period, and the calorie content of each one was determined. (This determination entails destroying the product, so a census would certainly not be desirable!) Here are the resulting observations:
255244239242265245259248225226251233
Does the data suggest that the true average calorie content differs from the stated calorie content of 240?
Construct a 95% confidence interval for the true average calorie content for this type of frozen dinner. Is this interval consistent with the results of your test of hypothesis above? Explain.
III.t-Test of Hypothesis and Confidence Intervalin a Matched Pairs Design
Example #5: In a study of memory recall, eight students from a largepsychology class were selected at random and given 10 minutes to memorize a list of 20 nonsense words. Each was asked to list as many of the words as he or she could remember both 1 hour and 24 hours later, as shown in the accompanying table. Is there evidence to suggest that the mean number of words recalled after 1 hour exceeds the mean recall after 24 hours by more than 3? Use a level 0.01 test.
Subject:12345678
1 hr. later14121871191615
24 hrs. later104146961212
Differences
Recall: In this example, 2 measurements (recall 1 hour later and 24 hours later)are
taken for each subject. Hence it is called a matched pairs design.
In addition, the two scores are condensed into one by considering the
difference between the scores.
Example #6: The humorous paper “Will Humans Swim Faster or Slower in Syrup?” (American Institute of Chemical Engineers Journal [2004]: 2646-2647) investigates the fluid mechanics of swimming. Twenty swimmers each swam a specified distance in a water-filled pool and in a pool where the water was thickened with food grade guar gum to create a syrup-like consistency. Velocity, in meters per second, was recorded. The authors of the paper concluded that swimming in syrup does not change swimming speed. Are the given data consistent with this conclusion? Carry out a hypothesis test using a .01 significance level.
Subject:12345678910
Water:0.900.921.001.101.201.251.251.301.351.40
Syrup:0.920.960.951.131.221.201.261.301.341.41
Subject:11121314151617181920
Water:1.401.501.651.701.751.801.801.851.901.95
Syrup:1.441.521.581.701.801.761.841.891.881.95
Using the swimming data above, determine a 99% confidence interval for the true
mean of the population of differences between water- and syrup-swimming speeds.
Is this interval estimate consistent with your test of hypothesis results above?
IV.Warning Label for using the Student’s t -Distribution
In general, t procedures are quite robust against lack of normality, but are
strongly influenced by outliers.
A confidence interval or test of hypothesis is called robust if the confidence interval or P-value does not change very much when the assumptions of the procedure are violated.
Conditions:
- The assumption that the data are a SRS from the population is more important than the assumption that the population distribution is normal.
- If the sample size is very small (n < 15), use the t procedures only if the data are close to normal. If the data are clearly not normal or if outliers are present, do not use t.
- If the sample size is not very small (15 < n < 30), use the t procedures except in the presence of outliers or strong skewness.
- If the sample size is large (n > 30), the t procedures can be used even in the case of strongly skewed distributions. Outliers should always be examined!