10-2 Comparing Means 3/28/2018

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

The Mean and St. Dev

Scenario 1

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?

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

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

Confidence Interval using T

North-South (23.7-34.53) ± (1.699)(4.12) -10.83 ± 7 -17.83 -- -3.83 We are 90% confident that the diameter of trees in the north is between 3.83 and 17.83 cm smaller than the diameter of trees in the south

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.

1.2 ± (1.697)(.418) 1.2 ± .709 .491- 1.909 We are 90% confident that county #1 has between .491 and 1.909 more trees per acre than county #2

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

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?

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

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

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?

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

Tcdf(1.043,BIG, 69) ==.15 (x2) p-value =.3 𝐻 0 : µ 1 = µ 2 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 ((102-100)-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

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?

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?

P-value: .0000235 Yes, the realtor would conclude that housing prices have increased

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