1. 10-2 Comparing Means
3/28/2018

2. Comparing Means
This is the exact same thing that we did last class with proportions
We have two samples (each from a different population)
We want to be able to draw conclusions about the DIFFERENCE between the two POPULATIONS
Based on the samples

3. The Mean and St. Dev

4. Scenario 1

5. First type of problem
The heights (in inches) of 10-year-old girls follows a normal distribution N(56.4, 2.7), and the heights of 10-year-old boys (in inches) follows a normal distribution N(55.7, 3.8). A researcher takes an SRS of 12 girls and an SRS of 8 boys. The researcher reports that the heights of 10-year-old boys and girls is the same, based on the sample.
Find the probability of getting a result as extreme as the one the researcher got
Based on the probability in Part A, do we have any reason to doubt the researcher’s claim?

6. Here we know the population standard deviation, so we can still use Z, or normal, distribution
Z=((value)-(mean))/(st dev)
Z=((0)-(.7))/1.55
Z=-.45
Normalcdf(-BIG, -.45, 0, 1)= .326
33% chance of getting a result as extreme—no reason to doubt the researchers

7. T-statistic
Usually we don’t know the population values, so we must use t distribution instead of normal

8. Confidence Interval using T

10. North-South
( ) ± (1.699)(4.12)
± 7
We are 90% confident that the diameter of trees in the north is between 3.83 and cm smaller than the diameter of trees in the south

11. Researchers take a random sample of the number of trees per acre in two different counties. In county #1, they take a sample of 57 acres, and find a mean of 17.1 trees, and a standard deviation of 2 trees. In county #2, they sample 31 acres and they find an average of 15.9 trees, and a standard deviation of 1.8 trees. Find a 90% confidence interval for the difference in means between the two counties. Assume all conditions for inference are met.

12. 1.2 ± (1.697)(.418)
1.2 ± .709
We are 90% confident that county #1 has between .491 and more trees per acre than county #2

13. Significance test
Are the means of the two populations significantly different from each other?

14. Different from a paired test
Last chapter we did a “paired” t-test
The grocery store (express lane vs regular) example from last chapter
The painkiller drug example on the test
This is different
How?

15. Different from a paired test
This is different
How?
In that scenario we had individual observations that received two separate treatments
We subtracted one treatment from the other, but it was using the same individual
Now the same individuals are not in both samples—each sample is randomly drawn from the given population, and the two samples are taken independently from each other

16. Difference Between Means
What we are doing now is called:
“Difference-in-means test”
“Difference between means test”
“Two-sample T-test”

18. A sample of 100 students in 1 school district revealed an average IQ of 102 with a standard deviation of 15
A sample of 70 students in a different school district revealed an average IQ of 100 with a standard deviation of 10
Is there a statistically significant difference between them?

19. Remember the 6 steps
A sample of 100 students in 1 school district revealed an average IQ of 102 with a standard deviation of 15
A sample of 70 students in a different school district revealed an average IQ of 100 with a standard deviation of 10
Is there a statistically significant difference between them?
Hypotheses
Sample
Conditions
Test Statistic
P-value
Conclusions

20. Tcdf(1.043,BIG, 69) ==.15 (x2) p-value =.3
𝐻 0 : µ 1 = µ 𝐻 𝑎 : µ 1 ≠ µ 2
Done
𝑛 1 >30, 𝑛 2 >30, As long as the districts are not tiny (less than 1,000 students), then 10% condition is also met
(( )-0)/ (1.918)
2/1.918
T=1.043
Tcdf(1.043,BIG, 69) ==.15 (x2) p-value =.3
Fail to reject null—cannot conclude that there is a difference between the two districts

21. My recommendation is to use the function on your calculator
Try 2-sampTtest on calc
Choose “no” for pooled
P-value is .299
Slight difference comes from df
My recommendation is to use the function on your calculator
Just remember to show work
A sample of 100 students in 1 school district revealed an average IQ of 102 with a standard deviation of 15
A sample of 70 students in a different school district revealed an average IQ of 100 with a standard deviation of 10
Is there a statistically significant difference between them?

22. You try
A realtor wants to know if housing prices are significantly higher in 2017 than they were a year before. He takes a random sample of 31 house sales in March 2017 and finds a mean price of $425,000 (standard deviation of $60,123). He takes a random sample of 32 house sales in March 2016 and finds a mean price of $363,000 (standard deviation of $51,363). Can the realtor conclude that housing prices were higher in 2017 than in 2016?

23. P-value:
Yes, the realtor would conclude that housing prices have increased

24. HW
Page 652: 40, 44, 46, 48, 51, 53, 55, 59, 61-64, 66, 68-70