Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
An explanation of hypothesis testing and t-tests using an example. The example involves testing whether the mean weight of all blocks is equal to a specified value under the null hypothesis, and calculating the p-value and confidence interval using a one-sample t test. The document also discusses the difference between the normal distribution and t-distribution and how to find areas under the t-distribution using table c or minitab.
Typology: Study notes
1 / 32
This page cannot be seen from the preview
Don't miss anything!
Christopher Holloman, The Ohio State University, Summer 2006
Christopher Holloman, The Ohio State University, Summer 2006
3
Intro. to Hypothesis Tests
Two of the most common types of statistical inference:
Goal is to estimate (and communicate uncertainty in our estimate of) a population parameter.
Goal is to assess the evidence provided by the data about some claim concerning the population.
Christopher Holloman, The Ohio State University, Summer 2006
Heads - Tom pays
Tails - Heather pays
Christopher Holloman, The Ohio State University, Summer 2006
Moral of the story: an outcome that
would rarely happen if a claim were
true is good evidence that the claim
is in fact not true.
This is the idea behind Hypothesis
Testing.
Christopher Holloman, The Ohio State University, Summer 2006
A hypothesis is a statement about the parameters in a population; we will be making statements about μ in Section 6.2.
A hypothesis test (or significance test) is a formal procedure for comparing observed data with a hypothesis whose truth we want to assess.
The results of a test are expressed in terms of a probability that measures how well the data and hypothesis agree.
Christopher Holloman, The Ohio State University, Summer 2006
Performing a Hypothesis Test
The null hypothesis (H 0 ) is the statement being tested. This is assumed “true” and compared to the data to see if there is evidence against it. A null hypothesis that we will see often is that the mean μ is equal to some standard value. Usually, null hypotheses give a statement of “no difference” or “no effect.”
Christopher Holloman, The Ohio State University, Summer 2006
Suppose we want to test the null
hypothesis that μ is some specified value, say μ 0. Then
H 0 : μ = μ 0
Note: We will always express H 0
using an equality sign.
Christopher Holloman, The Ohio State University, Summer 2006
Christopher Holloman, The Ohio State University, Summer 2006
Christopher Holloman, The Ohio State University, Summer 2006
In the strawberry bar example, this means:
“Is the sample average much less than we would expect to see if μ really is 16?”
We need to find the probability that we get a sample of 20 bars whose mean is 15.6 or less given that μ = 16.
Question: Why 15.6 or less? This is more extreme evidence against our null hypothesis and in support of the alterative hypothesis.
Christopher Holloman, The Ohio State University, Summer 2006
x
Christopher Holloman, The Ohio State University, Summer 2006
P-values in terms of the test statistic:
where z is the observed value of the test statistic and the probabilities are found using the standard normal distribution given in Table A.
H (^) a P - v a l u e A r e a u n d e r c u r v e
μμμμ < μμμμ (^) 0 P ( Z ≤≤≤≤ z )
μμμμ >^ μμμμ (^) 0 P ( Z^ ≥≥≥≥ z )
μμμμ ≠≠≠≠ μμμμ (^) 0 2 P ( Z (^) ≥≥≥≥ | z | )
Christopher Holloman, The Ohio State University, Summer 2006
A p-value is exact if the population
distribution is normal.
If the population is not normal, the
p-value approximates the true
probability for large n because of
the Central Limit Theorem.
Christopher Holloman, The Ohio State University, Summer 2006
The final step is to decide if there is a strong amount of evidence to reject H 0 in favor of Ha. This is accomplished using the P-value.
In our example, we got a P-value =0.0052. What does this tell us?
If H 0 is true (i.e., true mean weight is 16 oz), then the chance of getting a sample whose mean weight is 15.6 oz or less is 0.52%
Christopher Holloman, The Ohio State University, Summer 2006
Does it give evidence against H 0?
Yes, it is very unlikely that we
would observe a sample mean as
low as we did if H 0 is true.
Conclusion: We reject the null
hypothesis.
Christopher Holloman, The Ohio State University, Summer 2006
If we use an α level of 0.01 and the p-value is smaller than α, we can say that there is a less than one in one-hundred chance that we would observe data as extreme or more extreme than what we saw if the null hypothesis is true.
If the P-value ≤ α we say the data are statistically significant at level αααα.
Note: When we do not reject H 0 , we are not claiming H 0 is true. We are just concluding there is not sufficient evidence to reject it.
Christopher Holloman, The Ohio State University, Summer 2006
Example: Ameritech
Suppose last year Ameritech’s repair service took an average of 3. days to fix customer complaints. One of the managers is assigned to check if this year’s data show a different average time to fix problems. He collects a random sample of 30 customer complaints and finds that the average time taken to fix them is 2.1 days. Assume that the standard deviation of the time taken to fix the complaints is 2.5 days. Is this good evidence that the average time taken to fix the complaints is not 3.1 days?
Christopher Holloman, The Ohio State University, Summer 2006
30
0 2.^13.^1 =−
n
x z σ
μ
Christopher Holloman, The Ohio State University, Summer 2006
Calculate P-value: Since this is a two-sided test, we find the area to the left of -2.19 and then double it to get the p-value.
P-value = 2 x 0.0143 = 0.
What is your conclusion at the 0.05 level? Since the p-value is less than 0.05, the test is significant at the 0.05 level. We reject the null.
What is your conclusion at the 0.01 level? The p- value is larger than 0.01, so the test is not significant at this level. We would not reject the null at this level.
Christopher Holloman, The Ohio State University, Summer 2006
Christopher Holloman, The Ohio State University, Summer 2006
Tests from Confidence Intervals
We have covered two types of statistical inference procedures for the population mean μ:
Confidence Interval (CI) and Tests of Significance
Question: Is there any relationship between hypothesis tests and confidence intervals? Answer: Yes, a level α two-sided test rejects a hypothesis H 0 :μ = μ 0 exactly when the value μ 0 falls outside a level (1-α) CI for μ.
Christopher Holloman, The Ohio State University, Summer 2006
Example: Concrete Block
Bud’s Home Center sells concrete blocks. Bud wants to estimate the average weight of all blocks in stock. A sample of 64 blocks has a mean weight of 65.5 lbs. Assume that the weights of blocks vary with standard deviation 4.6 lbs.
Christopher Holloman, The Ohio State University, Summer 2006
Christopher Holloman, The Ohio State University, Summer 2006
Christopher Holloman, The Ohio State University, Summer 2006
Christopher Holloman, The Ohio State University, Summer 2006
40
Confidence Intervals and Hypothesis Tests in Minitab
Note: This function is for computing confidence intervals and hypothesis tests of μ, the population mean, assuming the population standard deviation is known. (Section 6.1 and Section 6.2)
