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Chapter 11.1 Inference for the Mean of a

Population.

Example 1 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 at right.

Employee Time

1 66

2 70

3 75

4 88

5 69

6 71

7 71

8 63

9 89

10 86

When the standard deviation of a statistic is

estimated from the data, the result is called the

standard error of the statistic, and is given by

s/vn.

When we use this estimator, the statistic that

results does not have a normal distribution,

instead it has a new distribution, called the

t-distribution.

Does the data provide evidence that the mean for

this company is greater than 72 minutes?

What is different about this problem?

Time for some Nspiration!

One-Sample z-statistic

- s known

m

z

One-sample t-statistic

- s unknown

m

t

The variability of the t-statistic is controlled

by the Sample Size. The number of degrees of

freeom is equal to n-1 .

- ASSUMING NORMALITY?
- SRS is extremely important.
- Check for skewness.
- Check for outliers.
- If necessary, make a cautionary statement.
- In Real-Life, statisticians and researchers try

very hard to avoid small samples.

Use a Box and Whisker to check.

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 43 46 27 19 47 19 2

6 35 34 15 44 40 38 31 46 52 25 35 35 33 29 34 41

49 28 52 47 35 48 22 33 41 51 27 14 54 45 At the

a .1, is there sufficient evidence to suggest

that this districts third graders reading

ability is different than the national mean of 34?

SRS?

- I have an SRS of third-graders

Normal? How do you know?

- Since the sample size is large, the sampling

distribution is approximately normally

distributed - OR
- Since the histogram is unimodal with no outliers,

the sampling distribution is approximately

normally distributed

Name the Test!! One Sample t-test for mean

Do you know s?

What are your hypothesis statements? Is there a

key word?

- s is unknown

Plug values into formula.

p-value tcdf(.6467,1E99,43).2606(2).5212

Use tcdf to calculate p-value.

a .1

Compare your p-value to a make decision

Conclusion

Since p-value gt a, I fail to reject the null

hypothesis.

There is not sufficient evidence to suggest that

the true mean reading ability of the districts

third-graders is different than the national mean

of 34.

Write conclusion in context in terms of Ha.

Back to Example 1. The times are recorded

below. Employee 1 2 3 4 5 6

7 8 9 10 Time 66 70 75 88 69 71

71 63 89 86 Does this data provide evidence

that the mean for this company is greater than 72

minutes?

SRS?

- I have an SRS of employees

- Since the histogram has no outliers and is

roughly symmetric, the sampling distribution is

approximately normally distributed

Normal? How do you know?

Do you know s?

What are your hypothesis statements? Is there a

key word?

- s is unknown, therefore we are using a 1 sample

t-test

H0 m 72 where m is the true of min spent on

PT Ha m 72 time spent by this companys

employees

Use tcdf to calculate p-value.

Plug values into formula.

p-value tcdf(.937,1E99,9).1866(2).3732

Compare your p-value to a make decision

Conclusion

Since p-value gt 15, I fail to reject the null

hypothesis that this companys employees spend 72

minutes on average on Personal Technology uses.

There is not sufficient evidence to suggest that

the true amount of time spent on personal

technology use by employees of this company is

more than the national mean of 72 min.

Write conclusion in context in terms of Ha.

Now for the fun calculator stuff!

Example 3 The Wall Street Journal (January 27,

1994) reported that based on sales in a chain of

Midwestern grocery stores, Presidents 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?

- Assume
- Have an SRS of weeks
- Distribution of sales is approximately normal due

to large sample size - s unknown
- H0 m 1323 where m is the true mean cookie

sales - Ha m ? 1323 per week
- Since p-value lt a of 0.05, I reject the null

hypothesis. There is sufficient to suggest that

the sales of cookies are different from the

earlier figure.

Name the Test!! One Sample t-test for mean

- Example 3 Presidents 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. Compute a 95

confidence interval for the mean weekly sales

rate. - CI (1105.30, 1310.70)
- Based on this interval, is the mean weekly sales

rate statistically different from the reported

1323?

What do you notice about the decision from the

confidence interval the hypothesis test?

Remember your, p-value .01475 At a .02, we

would reject H0.

- What decision would you make on Example 3 if a

.01? - What confidence level would be correct to use?
- Does that confidence interval provide the same

decision? - If Ha m lt 1323, what decision would the

hypothesis test give at a .02? - Now, what confidence level is appropriate for

this alternative hypothesis?

A 96 CI (1100, 1316). Since 1323 is not in

the interval, we would reject H0.

You would fail to reject H0 since the p-value gt a.

You should use a 99 confidence level for a

two-sided hypothesis test at a .01.

The 98 CI (1084.40, 1331.60) - Since 1323

is in the interval, we would fail to reject

H0. Why are we getting different answers?

Tail probabilities between the significant level

(a) and the confidence level MUST match!)

In a CI, the tails have equal area so there

should also be 2 in the upper tail

CI (1068.6 , 1346.40) - Since 1323 is in

this interval we would fail to reject H0.

a .02

.02

.96

That leaves 96 in the middle that should be

your confidence level

Ex4 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 Test

- A special type of
- t-inference

Matched Pairs two forms

- Pair individuals by certain characteristics
- Randomly select treatment for individual A
- Individual B is assigned to other treatment
- Assignment of B is dependent on assignment of A

- Individual persons or items receive both

treatments - Order of treatments are randomly assigned or

before after measurements are taken - The two measures are dependent on the individual

Is this an example of matched pairs?

- 1)A college wants to see if theres a difference

in time it took last years 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

No, there is no pairing of individuals, you have

two independent samples

Is this an example of matched pairs?

- 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

No, there is no pairing of individuals, you have

two independent samples If you would have the

same people taste both brands in random order,

then it would be an example of matched pairs.

Is this an example of matched pairs?

- 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 persons weight is

measured again.

Yes, you have two measurements that are dependent

on each individual.

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.)

You may subtract either way just be careful

when writing Ha

Day 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Morning 8 9 7 9 10 13 10 8 2 5 7 7 6 8 7

After-noon 8 10 9 8 9 11 8 10 4 7 8 9 6 6 9

Since you have two values for each day, they are

dependent on the day making this data matched

pairs

First, you must find the differences for each day.

Day 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Morning 8 9 7 9 10 13 10 8 2 5 7 7 6 8 7

After-noon 8 10 9 8 9 11 8 10 4 7 8 9 6 6 9

Differences 0 -1 -2 1 1 2 2 -2 -2 -2 -1 -2 0 2 -2

I subtracted Morning afternoon You could

subtract the other way!

- Assumptions
- Have an SRS of days for whale-watching
- s unknown
- Since the boxplot doesnt show any outliers, we

can assume the distribution is approximately

normal.

You need to state assumptions using the

differences!

Notice the skewness of the boxplot, however, with

no outliers, we can still assume normality!

Differences 0 -1 -2 1 1 2 2 -2 -2 -2 -1 -2 0 2 -2

Is there sufficient evidence that more whales are

sighted in the afternoon?

Be careful writing your Ha! Think about how you

subtracted M-A If afternoon is more should the

differences be or -? Dont look at numbers!!!!

If you subtract afternoon morning then Ha mDgt0

H0 mD 0 Ha mD lt 0 Where mD is the true mean

difference in whale sightings from morning minus

afternoon

Notice we used mD for differences it equals 0

since the null should be that there is NO

difference.

Differences 0 -1 -2 1 1 2 2 -2 -2 -2 -1 -2 0 2 -2

finishing the hypothesis test Since p-value

gt a, I fail to reject H0. There is insufficient

evidence to suggest that more whales are sighted

in the afternoon than in the morning.

In your calculator, perform a t-test using the

differences (L3)

Notice that if you subtracted A-M, then your test

statistic t .945, but p-value would be the

same

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.

Player Before After

1 13 18

2 20 37

3 17 40

4 13 35

5 13 30

6 16 20

7 15 33

8 16 19

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?

Player Before After

1 13 18

2 20 37

3 17 40

4 13 35

5 13 30

6 16 20

7 15 33

8 16 19