Download Comparing Two Means - Introduction to Statistics - Lecture notes and more Study notes Statistics in PDF only on Docsity!
Comparing Two of Something I. Means Comparing two means: o Tires for tread wear o Cough syrup effectiveness (any two drugs) o Average age of students at different schools Large samples n ≥ 30 or σ known
Take all samples from A and their means: X^ 1A , X^ 1B ,....
Subtract the means of all samples from B: X^ 2A , X^ 2B ,....
This gives a distribution of ( X^ 1 - X^ 2 ) values.
Our data are ( X^ 1 - X^ 2 )
If the populations have the same average (μ 1 = μ 2 ) then the values will usually be 0. If not our expected value for a given sample is μ 1 - μ 2
Since we are taking all values including extremes, a few ( X^ 1 - X^ 2 ) values will be
large. Sample 1A n ≥ 30 X (^) 1A Sample 2A n ≥ 30 X (^) 2A Population 1 Brand A
Population 2 Brand B
Sample 2A n ≥ 30 X (^) 2B Sample 1B n ≥ 30 X (^) 1B
With n ≥ 30 we get a normal distribution of mean differences. Using our test value format: Test value = (observed value) - (expected value) standard error So we get a new formula for calculating Test values. Test value for comparing two means with large sample sizes or known σ 1 and σ 2.
z =
( X 1 − X 2 ) − ( μ 1 − μ 2 )
2
n 1
2
n 2
Note: This is if we know σ 1 and σ 2. If not we have large samples and we use s 1 and s 2 (Central Limit Theorem) Note in testing means the null hypothesis is always the same: H 0 : μ 1 = μ 2 So for our purposes μ 1 - μ 2 = 0
Example 2 : A medical researcher wishes to see whether the pulse rates of smokers are higher than the pulse rates of non-smokers. Samples of 100 of each were tested with the following results: Smokers Non-smokers
X 1 = 90 X 2 = 88
s 1 = 5 s 2 = 6 n 1 = 100 n 2 = 100 Can the researcher conclude at α = 0.05 that smokers have higher pulse rates? We can also calculate the confidence interval for the expected difference between the two populations: μ 1 - μ 2 ( X (^) 1 - X (^) 2 ) - zα/ σ 1 2 n 1
σ (^2) 2 n 2 < μ 1 - μ 2 < ( X (^) 1 - X (^) 2 ) + zα/ σ 1 2 n 1
σ (^2) 2 n 2 Again if we don’t know σ 1 and σ 2 we need n 1 and n 2 large enough to use s 1 and s 2.
Example: A survey of 50 hotels in each of two cities found the average hotel rate in Monterey to be $88.42 with standard deviation $5.62 and the average room rate in Salinas to be $80.61 with standard deviation $4.83. Find the 95% confidence interval for the difference between the mean hotel prices in the two cities.
Example: The average farm size in Monterey County is 156 acres. The average farm size in Humbolt County is 162 acres. Assume the data values were obtained from 2 samples with standard deviations 32 and 11 acres respectively and sample sized of 8 and 10 respectively. Find the 95% Confidence interval for the differences in the acreages and compare it to our previous results for hypothesis testing.
Example: Researchers found that 12 of 34 small nursing homes had a vaccination rate of less than 80%, while 17 out of 24 large nursing homes had a vaccination rate of less than 80%. At a 0.05 level of significance, test the claim that there is no difference in the proportions of the small and large nursing homes with a resident vaccination rate of less than 80%. (You Try)Example: In a sample of 200 workers, 45% said that they missed work because of personal illness. Ten years ago in a sample of 200 workers, 35% said they missed work because of personal illness. With .01 significance, is there a difference in the proportion?
