Two independent samples

Difference of Means

CONDITIONS:1) The two samples are chosen independently of each

other. OR The treatments are randomly assigned to individuals

or objects or (vice versa) 2) The sample distributions for both samples should be

approximately normal- the populations are known to be normal, or

- the sample sizes are large (n 30), or - graph data to show approximately normal3) 10% rule – Both samples should be less than 10% of

their respective populations

Differences of Means (Using Independent Samples)

Confidence intervals:

statistic of SD valuecritical statisticCI

21xx *t

2

2

2

1

2

1

n

s

n

s

Called standard

error

Differences of Means (Using Independent Samples)

Degrees of FreedomOption 1: use the smaller of the

two values n1 – 1 and n2 – 1

This will produce conservative results – higher p-values & lower confidence.

Option 2: approximation used by technology

2

2

2

21

2

1

1

2

2

2

2

1

2

1

11

11

ns

nns

n

ns

ns

df

Calculator does this automatically!

Hypothesis Statements:

H0: m1 - m2 = hypothesized value

Ha: m1 - m2 < hypothesized value

Ha: m1 - m2 > hypothesized value

Ha: m1 - m2 ≠ hypothesized value

H0: m1 = m2

Ha: m1< m2

Ha: m1> m2

Ha: m1 ≠ m2

Differences of Means (Using Independent Samples)

Hypothesis Test:

statistic of SD

parameter - statisticstatisticTest

t

2

2

2

1

2

1

2121

ns

nsxx

State the degrees of freedom

Differences of Means (Using Independent Samples)

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 B18.9 7.5 12

Example 1

Is there sufficient evidence that these drugs differ in the speed at which they enter the blood stream?

Parameters and Hypotheses

μA = the true mean absorption time in minutes for brand A

Assumptions (Conditions)

Since the conditions are met, a t-test for the two-sample means is appropriate.

1) The samples must be random and independent which is stated in the problem.

H0: μA - μB = 0Ha: μA - μB 0

3) The samples should be less than 10% of their populations. The population should be at least 240 people, which I will assume.

2) The sample distributions should be approximately normal. Since it is stated in the problem that the population is normal then the sample distributions are aprroximately normal.

4) A and B are both unknown

μB = the true mean absorption time in minutes for brand BμA - μB = the true difference in means in absorption times in minutes for brands A and B

2 ( .3619) .721p value P t

Calculations

.721 .05

20.1Ax

12An

= 0.05

8.7As

0A B

.361921df

18.9Bx

7.5Bs

12Bn

2 2

A B A B

A B

A B

x xt

s s

n n

2 2

20.1 18.9 0

8.7 7.512 12

Conclusion:

Decision: Since p-value > a, I fail to reject the null hypothesis at the .05 level.

There is not sufficient evidence to suggest that there is a difference in the true mean absorption time in minutes for Brand A and Brand B.

#5 Commuting. A man who moves to a new city sees that there are two routes he could take to work. A neighbor who has lived there a long time tells him Route A will average 5 minutes faster than Route B. The man decides to experiment. Each day he flips a coin to determine which way to go, driving each route 20 days. He finds that Route A takes an average of 40 minutes, with a standard deviation of 3 minutes, and Route B takes an average of 43 minutes, with a standard deviation of 2 minutes. Histograms of travel times for the routes are roughly symmetric and show no outliers.

a) Find a 95% confidence interval for the difference in average commuting time for the two routes.

b) Should the man believe the old-timer’s claim that he can save an average of 5 minutes a day by always driving Route A? Explain.

Page 567: #5

μA = the true mean time it takes to commute taking Route A

State the parameters

Justify the confidence interval needed (state assumptions)

3) The samples should be less than 10% of the population. The population should be at least 400 days, which I will assume.

2) The sample distributions should be approximately normal. It is stated in the problem that graphs of the travel times are roughly symmetric and show no outliers, so we will assume the distributions are approximately normal.

Since the conditions are satisfied a t – interval for the difference of means is appropriate.

1) The samples must be random and independent which is state in the problem.

4) A and B are both unknown

μB = the true mean time it takes to commute taking Route B

μB - μA = the true difference in means in time it takes to commute taking Route B from Route A

We are 95% confident that the true mean difference between the commute times is between 1.3599 and 4.6401 minutes.

Calculate the confidence interval.

95% CI

Explain the interval in the context of the problem.

40Ax

3As

1.3599, 4.6401 33df

43Bx

2Bs

20Bn 20An

2 2B A

B AB A

s sx x t

n n

2 22 343 40

20 20 2.043