Advanced Math Topics

8.3-8.5 Sample Means

The Food and Drug Administration is inspecting a tobacco company for tar content.They randomly select 6 different boxes that each have 100 cigarettes and tests them.The average tar content of each box (mg content per cigarette) is shown below.

15.8 16.2 14.8 15.8 15.3 13.9

Find the mean and standard deviation.

x 15.8 + 16.2 + 14.8 + 15.8 + 15.3 + 13.9 6

This is read as “mu sub x bar” because it is the mean of the means.

= 15.3

The Food and Drug Administration is inspecting a tobacco company for tar content.They randomly select 6 different boxes that each have 100 cigarettes and tests them.The average tar content of each box (mg content per cigarette) is shown below.

15.8 16.2 14.8 15.8 15.3 13.9

Find the mean and standard deviation.

x 15.3

Formula The standard deviation of the sample means is:

2)( xx xx x15.8

16.2

14.8

15.8

15.3

13.9

15.8 – 15.3 = 0.5 16.2 – 15.3 = 0.9 14.8 – 15.3 = -0.5 15.8 – 15.3 = 0.5 15.3 – 15.3 = 0.0 13.9 – 15.3 = -1.4

(0.5)2 = .25

(0.9)2 = .81

(-0.5)2 = .25

(0.5)2 = .25

(0.0)2 = 0

(-1.4)2 = 1.96

Sum = 3.52

σ = .7659

n is the # ofsample means

n

x x

2)(

6

52.3

The distribution of the sample means is approximately normally distributed.

But can you follow this?

From our previous example, we had a sample size of 100 cigarettes and the sample mean was 15.3.

The distribution would look something like this.

μx = 15.3

What if we had sample sizes of 50(instead of 100)?

Likely, what would the sampling distribution mean be?15.3, the same.

Would the bell curve look the same?The smaller the sample size, it is more likely to have a sample mean that isan outlier. Thus, the bell curve would be more spread out.

μx = 15.3

Thus, the standard deviation would be larger.

In summary, as the size of the sample gets smaller,the standard deviation gets larger and the bell curve becomes more spread out.

From the HW P. 418

1) During each week of the first 6 weeks of the year, a doctor delivered 9, 10, 5, 8, 7and 6 babies.

9,10; 9,5; 9,8; 9,7; 9,6; 10,5;…..7,6 There are 15 total samples.

a) Make a list of all possible samples of size 2 that can be made from the list.

From the HW P. 418

1) During each week of the first 6 weeks of the year, a doctor delivered 9, 10, 5, 8, 7and 6 babies.

The sample means are 9.5, 7, 8.5, 8, 7.5, 7.5, 9, 8.5, 8, 6.5, 6, 5.5, 7.5, 7, 6.5.

b) Determine the mean of each of these samples and form a sampling distribution ofthese sample means.

From the HW P. 418

1) During each week of the first 6 weeks of the year, a doctor delivered 9, 10, 5, 8, 7and 6 babies.

7.5

c) Determine the mean of the sampling distribution.

From the HW P. 418

1) During each week of the first 6 weeks of the year, a doctor delivered 9, 10, 5, 8, 7and 6 babies.

1.0801

d) Determine the standard deviation of the sampling distribution.

HW

P. 419 #1, 2, 4(test-type question), 5a,b,cExplain why your two standard deviations are different in #5