Warm-Up Jeremy, as a point guard on the HIS junior varsity

basketball team, connected on 54% of his shots from beyond the three-point line. In his first year on the varsity team, he connected on 73 of 120 three-point attempts.

a) Is there sufficient evidence that he is shooting better than he did as a junior varsity player?

b) Suppose Jeremy makes 15 three-point shots in the next two games. Could the outcome of these shots change your conclusion above?

Some studies have shown that lean and obese people spend their time differently. Obese people spend fewer minutes per day standing and walking than do lean people who are similar in age, overall health, and occupation. Is this difference biological or is it a response to obesity (people become less active when they gain weight?). A small pilot study looked at this. The subjects were 7 mildly obese people who were healthy and didn’t exercise. The subjects volunteered to participate in a weight-loss program for 8 weeks during which they lost an average of 8 kgs (17.6 lbs). Before and after weight loss, subjects wore monitors that recorded their movement for 10 days. The table shows the minutes per day spent standing and walking. Carry out a hypothesis test to find out if mildly obese people increase the time they spend standing and walking when they lose weight.

***matched pairs, σ unknown=> t-procedure

Subject Before After

1 293 264

2 330 335

3 353 387

4 354 307

5 400 387

6 454 358

7 552

549

13.1: Comparing 2 means

• Instead of “mean difference” comparing, we compare ___________________________.

• It is the most common statistical practice.

• Types of 2-sample problems:

1)

2)

Note:

• 2 samples can be ____________________(n1≠n2)

• Examine both samples ______________

• We measure the same variable for both samples

• 3 Assumptions for Comparing 2 Means: 1)

2)

3)

Notation to describe the 2 populations

• Population (1 and 2)

• Variables (x1, x2)

• Mean (μ1, μ2)

• Standard deviation (σ1, σ2)

• Sample size (n1, n2)

• Sample means (x-bar1, x-bar2)

• Sample standard deviations (s1, s2)

• We want to compare the 2 population means by:

– CI for μ1-μ2 (their differences)

– testing Ho: μ1=μ2

The sampling distribution of ______

• The _____________ of sample means is an unbiased estimator of the ____________ of population means

• The ____________ of the difference is the _______ of the variances of ____ and ____

• If the two population distributions are both _______, then the distribution of the sample means is ____________________________.

• If (n1+n2) > _____= Normal (CLT)

2-sample z- and t- test statistic • If we know ___ and if we know that ________ has a

___________________, we can standardize it to obtain a standard normal z statistic.

z =

If ___ known... • Z: says how far __________ is from its mean in

standard deviation units

• To test Ho: _______, solve ___________

• Use __________ • For degree of freedom, check both:

=> Use the smaller of the 2

If __ unknown...

• Because we don’t know___ and ___, we follow the pattern of:

• 2 Sample t Procedures for a Confidence Interval:

• 2 Sample t Procedures for a Hypothesis Test:

Example • Does increasing the amount of calcium in our diet reduce blood pressure?

Examination of a large sample revealed a relationship between calcium intake and blood pressure... The subjects in the experiment were 21 healthy black men. A randomly chosen group of 10 men received a calcium supplement for 12 weeks, and the control group of 11 men took a placebo….the response variable is the decrease in b.p. after 12 weeks; an increase appears as a negative response.

• Group 1 = calcium group

7 -4 18 17 -3 -5 1 10 11 -2 L1

• Group 2 = placebo group

-1 12 -1 -3 3 -5 5 2 -11 -1 -3 L2

1)Is this good evidence that calcium decreases blood pressure in the population more than a placebo does?

2) Find a 90% confidence interval for this prob.

Example cont’d: Group Treatment n x-bar S

1

2

P: μ = Fits a 2-Sample Setting: Let μ1 = Let μ2 = H: Ho: μ1=μ2 Ha: μ1>μ2

A:

Example cont’d:

T: Calc:

t =

p =

***Make sure to choose “No” for pooled

CI at 90%: Calc:

C:

In a study of heart surgery, one issue was the effect of drugs called beta-blockers on the pulse rate of patients during surgery. The available subjects were divided at random into two groups of 30 patients each. One group received a beta-blocker; the other group received a placebo. The pulse rate of each patient at a critical point during the operation was recorded. The treatment group had mean 65.2 and standard deviation 7.8. For the control group, the mean was 70.3 and the standard deviation was 8.3.

1. Perform an appropriate significance test to see if beta-blockers reduce the pulse rate.

2. Give a 99% confidence interval for the difference in mean pulse rates. Interpret the confidence interval you obtain.

LeRoy, a starting player for a major college basketball team, made only 40% of his free throws last season. During the summer, he worked on developing a softer shot in hopes of improving his free throw accuracy. In the first eight games of this season, LeRoy made 25 free throws in 40 attempts.

1. You want to investigate whether LeRoy’s work over the summer will result in a higher proportion of free-throw successes this season. What conclusion would you draw at the = 0.01 level about LeRoy’s free throw shooting? Justify your answer.

2. Construct and interpret a 90% confidence interval for the proportion of free throws LeRoy will make this season.

• In an experiment to study the effect of the spectrum of the ambient light on the growth of plants, researchers assigned tobacco seedlings at random to 2 groups of 8 plants each. The plants were grown in a greenhouse under identical conditions except for lighting. The experimental group was grown under blue light, the control group under natural light. The researchers want to know: Does blue light reduce the plant’s ability to grow?

• Here’s the data on stem growth: • Control group (natural): 4.3 3.9 4.1 3.8 4.2 4.2 4.2

4.1 • Experimental (blue): 3.1 3.2 2.7 3.0 2.9 3.2 2.9 3.1 • Construct and interpret a 90% Confidence

Interval for the difference in mean plant growth.

13.2: Comparing 2 Proportions • In a two-sample problem, we want to compare

__________________ or the responses to __________________ based on two independent samples.

• In these problems, we will compare ___________________________________________.

• Notation Population Proportion: Sample size: Sample Proportion:

• We compare the populations by doing inference about the

________________________________________________. The statistic that estimates this difference is the difference between the sample proportions ________________.

The sampling distribution of: _________

• The mean of _______ is _________. • The variance of _______ is:

• When the samples are _________, the

distribution is _____________________. The standard deviation involves the known

parameters p1 and p2, so we ____________________________________ in order to do inference.

Confidence Interval for Comparing 2 Proportions

• Confidence Interval:

• Standard Error Equation:

• 3 Assumptions: 1) 2) 3)

Example • To study the long-term effects of preschool programs

for poor children, a program followed 2 groups of kids since childhood. One group (size 61) of Population 1 was poor kids with no preschool, while the other (size 62) of Population 2 was poor kids with preschool. In the past 10 years, 38 of the preschool sample and 49 of the control sample have needed welfare.

• Compute a 95% confidence interval measuring the effect of preschool in reducing the proportion of people who later need social services.

p-hat1 = _____, p-hat2 = _____

Population Description Sample Size Sample Proportion

1 Control: no preschool

2 With preschool

P:

H: Ho: Ha:

A:

T:

C:

Significance Tests for: ______ • An observed difference between 2 sample

proportions can reflect a difference in the populations, or it may just be due to chance variation in random sampling.

Significance tests help us:

• As usual, the Ho says:

_________________________________

while the Ha says: ________________________________

Significance Test 2 Props

To test Ho=_________, find the pooled proportion p-hat of successes in both samples combined We pool the 2 samples and use the overall sample proportion to estimate __________________. This is called the: ____________________________ p-hat =

Z =

3 options for Ha=

1)

2)

3)

3 Assumptions:

1)

2)

3)

Example

• High levels of cholesterol in the blood are associated with higher risk of heart attacks. Will using a drug to lower cholesterol reduce heart attacks? A study assigned 2 treatments: 2051 men took the drug gemfibrozil to reduce cholesterol, while 2030 men took a placebo. Within 5 years, 56 men in the gemfibrozil group and 84 men in the placebo group had heart attacks.

• Is the benefit of gemfibrozil statistically significant at the .01 level?

p-hat1 = ____, p-hat2 = _____ Population Description Sample Size Proportion

1 Gemfibrozil

2 Placebo

P:

H: Ho: Ha:

A:

T:

C:

A study of “adverse symptoms” in users of over-the-counter pain relief medications assigned subjects at random to one of two common pain relievers: acetaminophen and ibuprofen. In all, 650 subjects took acetaminophen, and 44 experienced some adverse symptom. Of the 347 subjects who took ibuprofen, 49 had an adverse symptom. We want to assess the evidence that the two pain relievers differ in the proportion of people who experience an adverse symptom.

1. Identify the populations and parameters of interest. Then state hypotheses.

2. State and verify conditions for carrying out a significance test.

3. Calculate the test statistic and the P-value. 4. What conclusion would you draw? 5. Construct and interpret a 90% confidence interval for the

size of the difference.

The table below gives the pretest and posttest scores on the MLA listening test in Spanish for 20 high school Spanish teachers who attended an intensive summer course in Spanish.

Teacher Pretest Posttest Teacher Pretest Posttest

1 30 29 11 30 32

2 28 30 12 29 28

3 31 32 13 31 34

4 26 30 14 29 32

5 20 16 15 34 32

6 30 25 16 20 27

7 34 31 17 26 28

8 15 18 18 25 29

9 28 33 19 31 32

10 20 25 20 29 32

1) We hope to show that attending the institute improves listening skills (matched pair!). Show a graphical check of the data (and report your conclusions on the validity of the test). Carry out a test of significance and include a complete conclusion at a 5% and 1% significance level. Include a 90% CI for the mean increase in score.

Extra Practice - proportions Mars Inc. claims they produce M&M’s with the following distribution: 30% brown, 20% red, 20% yellow, 10% orange, 10% green, 10% blue. A bag was randomly selected from the grocery store shelf, and the color counts were as follows: 16 brown, 11 red, 19 yellow, 5 orange, 7 green, 3 blue. Assuming that this is an SRS of all M&M’s, conduct an appropriate test of the manufacture’s claim for the proportion of yellows (show all work including proving assumptions). Calculate a 90% confidence interval.