t-distribution The Previously, for the sake of convenience, we have been assuming that when dealing with normal models for quantitative data that either , or , or both are known. In practice, this is never really the case. We have mentioned how the sample mean x is a good (unbiased) estimate of the population mean , but now we will also be using s, (the sample standard deviation), as a measure to estimate the population standard deviation, . Recall that:
1n s= (xi x)2 n 1 1
When is not known it is estimated from the sample standard deviation s, and the sampling distribution that results is said to follow a
Important properties of the t-distribution: 1. The t distribution is different for different sample sizes
2.
The t distribution has the same general symmetric bell shape as the normal distribution but it reflects the greater variability (with wider distributions) that is expected with small samples.
The t distribution has a mean of t = 0 (just as the standard normal distribution has a mean of z = 0). 4. As the sample size n gets larger, the t distribution gets closer to the normal distribution. 5.
3.
x t= s/ n
A (1 ) % confidence interval for (when is unknown) is defined as:
Example: The UCLA housing office wants to estimate the mean monthly rent for studios around the campus. A random sample of size n = 36 studios is selected from the area around UCLA. The sample mean is found to be x = $1200 and standard deviation of our sample s = $150. a. Construct a 95% confidence interval for the mean monthly rent of studios in the area around UCLA.
b. Construct a 99% confidence interval for the mean monthly rent of studios in the area around UCLA.
c. How do these intervals compare to the ones we created when we assumed was known?
In general for all confidence intervals, if everything else remains the same, then:
Statistical Inference Hypothesis Testing for Proportions Procedures for statistical inference Point estimation. Appropriate when the goal is to estimate a single best guess at the population parameter. Confidence interval. Appropriate when the goal is to estimate a likely range of values within which the population parameter lies. Hypotheses testing. Hypothesis: a statement about the parameters.
Concepts of Hypothesis Testing The critical concepts of hypothesis testing: 1. Define your null and alternative hypotheses Ho - the null hypothesis
Ha - the alternative hypothesis
It is important to decide upon Ha before performing any actual test
How to choose an appropriate alternative hypothesis (Ha) What determines the choice of an appropriate alternative hypothesis is what we know about the problem before we perform a test of statistical significance.
Example: A 1996 report from the U.S. Consumer Product Safety Commission claimed that at least 90% of all American homes have at least one smoke detector. A citys fire department has been running a public safety campaign about smoke detectors consisting of posters, billboards, and ads on radio and TV and in the newspaper. The city wonders if this concentrated effort has raised the local level above the 90% national rate.
Example: There are supposed to be 20% of orange M&Ms in any given bag. Suppose a bag of 122 M&Ms has 21 orange ones. Does this contradict the companys 20% claim?
A two-tail or two-sided test of a population proportion has the following null and alternative hypotheses:
A one-tail or one-sided test of a population proportion has the following null and alternative hypotheses:
2. Check all necessary assumptions Assumptions for qualitative data (proportions 1 sample)
3. Find the appropriate rejection region(s) based on the significance level (the size of the rejection region) and the alternative hypothesis (the direction(s) of the rejection region)
4. Compute the appropriate test statistic Test statistics for qualitative data (proportions 1 sample)
z= p p pq n
5. Find the p-value p-value:
extreme means far from what we would expect from H0.
6. State the appropriate conclusion from what the data are telling us We need to make a conclusion after carrying out the hypothesis test. What do we conclude? We can compare the P-value with a fixed value that we regard as decisive. This amounts to announcing in advance how much evidence against H0 we require in order to reject H0. The decisive value is called the significance level of the test. It is denoted by and the corresponding test is called a level test.
Example: A 1996 report from the U.S. Consumer Product Safety Commission claimed that at least 90% of all American homes have at least one smoke detector. A citys fire department has been running a public safety campaign about smoke detectors consisting of posters, billboards, and ads on radio and TV and in the newspaper. The city wonders if this concentrated effort has raised the local level above the 90% national rate. Building inspectors visit 400 randomly selected homes and find that 376 have smoke detectors. Is this strong evidence the that local rate is higher than the national rate? Assume = 0.05.
Example: There are supposed to be 20% of orange M&Ms in any given bag. Suppose a bag of 122 M&Ms has 21 orange ones. Does this contradict the companys 20% claim? Assume = 0.05.
Statistical Inference Hypothesis Testing for Means When performing a hypothesis test for quantitative data (means), the steps and interpretations remain the same as when testing a hypothesis for qualitative data (proportions) except that the test statistic now changes. 1. 2. State your hypotheses Check assumptions
3. 4.
Label your rejection region Compute your test statistic
5. 6.
Find the p-value (cant compute exact p-values from the t-table but you can compute an interval for the p-value) State your conclusion
Example: A health advocacy group tests whether the mean nicotine content of a brand of cigarettes is greater than the advertised value of 1.4 mg. The group takes a random sample of 30 cigarettes made by this specific brand and finds an average nicotine content of 1.8 mg with a standard deviation of 0.8 mg.
Check assumptions
What would our conclusion be if = 0.05? = 0.01?
Example: An aquarium needs to maintain a water temperature level of 40 degrees Fahrenheit in its penguin exhibit in order to keep the penguins happy. Any departure from this ideal temperature could prove disastrous. On each of 20 days, water temperature measurements were made in this exhibit and found an average water temperature of 39.2 degrees with a standard deviation 2 degrees. Are the penguins living in safe conditions?
What would our conclusion be if = 0.05? = 0.01?
Type I and Type II errors A Type I error
A Type II error
The probability of making a Type II error is labeled . Example H0: The person on trial for a crime is not guilty. (In the U.S., people are considered innocent unless proven otherwise.) Ha: The person on trial for a crime is guilty (The police believe this person is the main suspect.)
(They reject the null hypothesis even though the null hypothesis is actually true.)
(The jury fails to reject the null hypothesis even though the null hypothesis is false.)
Cigarette example: What would a type I and type II error be in the context of this problem? Recall: H0: = 1.4 mg Ha: > 1.4 mg Type I error:
Type II error:
Penguin example: What would a type I and type II error be in the context of this problem? Recall: H0: Ha:
= 40 temperature 40
temperature
Type I error:
Type II error:
Comparing Two Proportions Equipped with a basic understanding of confidence intervals and hypothesis tests, we now turn to the specific case of inference for the difference between two proportions. The approach, logic, and interpretations are the same. Only the assumptions for inference and our computation of the standard error change. Assumptions for inference:
Computing a confidence interval for the difference between two proportions
p1q1 p2q2 + p1 p2 z n1 n2
*
Example: Would being part of a support group that meets regularly help people who are wearing the nicotine patch quit smoking? A county health department tries an experiment using several hundred volunteers who were planning to use the patch to help them quit smoking. The subjects were randomly divided into two groups. People in Group 1 were given the patch and attended a weekly discussion meeting with counselors and others trying to quit. People in Group 2 also used the patch but did not participate in the counseling groups. After six months 46 of the 143 smokers in Group 1 and 30 of 151 smokers in Group 2 had successfully stopped smoking. Create a 95% confidence interval for the difference of smokers quitting between the two groups.
Performing a Hypothesis Test for a Difference Between Two Proportions H0: p1 = p2 Ha: p1 p2 OR p1 < p2 OR p1 > p2
p pooled
Success1 + Success2 = n1 + n2
p pooled q pooled p pooled q pooled + n1 n2
SE pooled ( p1 p2 ) =
p1 p2 z= SE pooled ( p1 p2 )
By performing a hypothesis test, do these results suggest that such support groups could be an effective way to help people stop smoking? Test at alpha = .05
Comparing Two Means The textbook gives a crazy formula for finding the degrees of freedom (df) when comparing two means. Dont use it. Just use df = n1 + n2 2 (# of obs. - # of groups) Assumptions for inference when comparing two means:
Form of a confidence interval:
s12 s22 y1 y2 tn* +n 2 n +n 2 1
1 2
Test statistic for a hypothesis test:
t=
y1 y 2 s 12 s 22 + n1 n2
Example: Resting pulse rates for a random sample of 26 smokers had a mean of 80 beats per minute and a standard deviation 5 bpm. Among 32 randomly chosen non-smokers, the mean and standard deviation were 74 and 6 bpm. Create a 95% confidence interval for the difference in mean pulse rates between the two groups.
Does this difference seem significant?
Perform a hypothesis test with alpha = .01 to examine if there is evidence of a difference in mean pulse rate between smokers and nonsmokers.