Name ______
AP STATISTICS CHAPTER 11: INFERENCE FOR A DISTRIBUTION
Chapter 11.1
Inference for the Mean of a Population
Ex: One concern employers have about the use of technology is the amount of time that employees spend each day making personal use of company technology, such as phone, e-mail, internet, and games. The Associated Press reports that, on average, workers spend 72 minutes a day on such personal technology uses. A CEO of a large company wants to know if the employees of her company are comparable to this survey. In a random sample of 10 employees, with the guarantee of anonymity, each reported their daily personal computer use. The times are recorded below.
Employee / Time1 / 66
2 / 70
3 / 75
4 / 88
5 / 69
6 / 71
7 / 71
8 / 63
9 / 89
10 / 86
Does this data provide evidence that the mean for this company is greater than 72 minutes?
What is different about this problem, compared to what we were given in chapter 10?
When the of a statistic is estimated from the data, the result is called the
of the statistic, and is given by .
When we use this estimator, the statistic that results does not have a normal distribution, instead it has a new distribution, called the .
One-sample z-statistic
One-sample t-statistic
The variability of the t-statistic is controlled by the .
The number of is equal to .
ASSUMING NORMALITY?
USINGTHE t PROCEDURES
1.
2.
3.
4.
Example 2: The Degree of Reading Power (DRP) is a test of the reading ability of children. Here are DRP scores for a random sample of 44 third-grade students in a suburban district:
40 26 39 14 42 18 25 54 48
43 46 27 19 47 19 26 45 22
35 34 15 44 40 38 31 14 33
46 52 25 35 35 33 29 27 41
34 41 49 28 52 47 35 51
At the α = .1, is there sufficient evidence to suggest that this district’s third graders reading ability is different than the national mean of 34?
Back to Example 1, Does this data provide evidence that the mean for this company is greater than 72 minutes?
Employee / Time1 / 66
2 / 70
3 / 75
4 / 88
5 / 69
6 / 71
7 / 71
8 / 63
9 / 89
10 / 86
Example 3: The Wall Street Journal (January 27, 1994) reported that based on sales in a chain of Midwestern grocery stores, President’s Choice Chocolate Chip Cookies were selling at a mean rate of $1323 per week. Suppose a random sample of 30 weeks in 1995 in the same stores showed that the cookies were selling at the average rate of $1208 with standard deviation of $275. Does this indicate that the sales of the cookies is different from the earlier figure?
Example 4: The times of first sprinkler activation (seconds) for a series of fire-prevention sprinklers were as follows:
27 41 22 27 23 35
30 33 24 27 28 22
24
Construct a 95% confidence interval for the mean activation time for the sprinklers.
MATCHED PAIRS PROCEDURES
Form 1:
Form 2:
Is this an example of matched pairs?
1)A college wants to see if there’s a difference in time it took last year’s class to find a job after graduation and the time it took the class from five years ago to find work after graduation. Researchers take a random sample from both classes and measure the number of days between graduation and first day of employment
2) In a taste test, a researcher asks people in a random sample to taste a certain brand of spring water and rate it. Another random sample of people is asked to taste a different brand of water and rate it. The researcher wants to compare these samples
3) A pharmaceutical company wants to test its new weight-loss drug. Before giving the drug to a random sample, company researchers take a weight measurement on each person. After a month of using the drug, each person’s weight is measured again.
Example: A whale-watching company noticed that many customers wanted to know whether it was better to book an excursion in the morning or the afternoon. To test this question, the company collected the following data on 15 randomly selected days over the past month. (Note: days were not consecutive.)
Player / Before / After1 / 13 / 18
2 / 20 / 37
3 / 17 / 40
4 / 13 / 35
5 / 13 / 30
6 / 16 / 20
7 / 15 / 33
8 / 16 / 19
Ex: The effect of exercise on the amount of lactic acid in the blood was examined in journal Research Quarterly for Exercise and Sport. Eight males were selected at random from those attending a week-long training camp. Blood lactate levels were measured before and after playing 3 games of racquetball, as shown in the table.
What is the parameter of interest in this problem?
Construct a 95% confidence interval for the mean
change in blood lactate level.
Based on the data, would you conclude that there is a significant difference, at the 5% level, that the mean difference in blood lactate level was over 10 points?
SUMMARY/QUESTIONS TO ASK IN CLASS
Name ______
AP STATISTICS CHAPTER 11: INFERENCE FOR A DISTRIBUTION
Two-Sample Inference Procedures with Means
• The goal of these inference procedures is to compare the responses to or to compare the characteristics of
• We have samples from each treatment or population
Formula for Mean:
Formula for Standard Deviation:
Example 1
Suppose we have a population of adult men with a mean height of 71 inches and standard deviation of 2.6 inches. We also have a population of adult women with a mean height of 65 inches and standard deviation of 2.3 inches. Assume heights are normally distributed.
a. Describe the distribution of the difference in heights between males and females (male-female).
b. What is the probability that the height of a randomly selected man is at most 5 inches taller than the height of a randomly selected woman?
c. What is the 70th percentile for the difference (male-female) in heights of a randomly selected man & woman?
d. What is the probability that the mean height of 30 men is at most 5 inches taller than the mean height of 30 women?
e. What is the 70th percentile for the difference (male-female) in mean heights of 30 men and 30 women?
Assumptions: (Use when)
Degrees of Freedom
Option 1:
Option 2:
Confidence Interval:
Hypothesis Test:
Example 2: Two competing headache remedies claim to give fast-acting relief. An experiment was performed to compare the mean lengths of time required for bodily absorption of brand A and brand B. Assume the absorption time is normally distributed. Twelve people were randomly selected and given an oral dosage of brand A. Another 12 were randomly selected and given an equal dosage of brand B. The length of time in minutes for the drugs to reach a specified level in the blood was recorded. The results follow:
mean SD n
Brand A 20.1 8.7 12
Brand B 18.9 7.5 12
Describe the shape & standard error for sampling distribution of the differences in the mean speed of absorption.
Find a 95% confidence interval difference in mean lengths of time required for bodily absorption of each brand.
Is there sufficient evidence that these drugs differ in the speed at which they enter the blood stream?
Suppose that the sample mean of Brand B is 16.5, then is Brand B faster?
Confidence interval statements:
Hypothesis Statements:
Example 3:
The length of time in minutes for the drugs to reach a specified level in the blood was recorded. The results follow:
mean SD n
Brand A 20.1 8.7 12
Brand B 18.9 7.5 12
Is there sufficient evidence that these drugs differ in the speed at which they enter the blood stream?
Example 4: An article in a professional journal examines the relationship between attitudes towards success at college-level mathematics. Twenty men and thirty-eight women selected at random from those identified at being high-risk of failure participated in the study. Each student was asked to respond to a series of questions, and the answers were combined to obtain a math anxiety score. For this particular scale, the higher the score, the lower the level of anxiety towards mathematics. Here are the summary values.
n / / sMales / 20 / 35.9 / 11.9
Females / 38 / 36.6 / 12.3
Does this data provide evidence that the mean anxiety score is different for women that it is for men?
Robustness:
Example 5: A modification has been made to the process for producing a certain type of time-zero film (film that begins to develop as soon as the picture is taken). Because the modification involves extra cost, it will be incorporated only if sample data indicate that the modification decreases true average development time by more than 1 second. Should the company incorporate the modification?
Original 8.6 5.1 4.5 5.4 6.3 6.6 5.7 8.5
Modified 5.5 4.0 3.8 6.0 5.8 4.9 7.0 5.7
SUMMARY/QUESTIONS TO ASK IN CLASS
Name ______
AP STATISTICS CHAPTER 11: INFERENCE FOR A DISTRIBUTION
SUMMARY OF INFERENCE PROCEDURES FOR DISTRIBUTIONS – t TESTS
SUMMARY/QUESTIONS TO ASK IN CLASS
Name ______
AP STATISTICS CHAPTER 11: INFERENCE FOR A DISTRIBUTION
ONE-SAMPLE t-TEST
Use when the sample is random and the population standard deviation is unknown. Verify normality, or the absence of outliers and strong skewness, based upon sample size.
ONE-SAMPLE t-TEST FOR DIFFERENCES
Use in a matched pairs setting, when population standard deviation is unknown. Verify normality, or the absence of outliers and strong skewness, based upon sample size.
STANDARD ERROR:
TEST STATISTIC:
CONFIDENCE INTERVAL:
DEGREES OF FREEDOM:
TWO-SAMPLE t-TEST
Use when
STANDARD ERROR:
TEST STATISTIC:
CONFIDENCE INTERVAL:
DEGREES OF FREEDOM:
SUMMARY/QUESTIONS TO ASK IN CLASS
Name ______
AP STATISTICS CHAPTER 11: INFERENCE FOR A DISTRIBUTION
SUMMARY/QUESTIONS TO ASK IN CLASS