Chapter 23 – Inferences About Means

Confidence Intervals & Hypothesis Tests We’ve spent the last few chapters working on creating confidence intervals and hypothesis tests for proportions. The basic concepts can also be applied to inferences about means with a couple of minor changes. The interpretations of these confidence intervals and hypothesis tests won’t be any different though.

Sampling Distribution Model Central Limit Theorem tells us that our sampling distribution model for means is Normal with: Mean = µ We can collect a random sample, but the problem with means is that we won’t know the true population standard deviation. Remember, for proportions, we could calculate our standard deviation using the population proportions.

Sample Standard Error We will have to estimate the population standard deviation σ with the sample standard deviation s. We use the Standard error:

Impact on our Analysis Since we didn’t know the population standard deviation, we have extra variation in our standard error because we had to use s. We will need to allow for this additional error so it doesn’t affect our margin of error and P-value. The shape of our model isn’t exactly Normal anymore.

Gosset’s (or Student’s) t William S. Gosset was an employee at Guinness Brewery in Dublin, Ireland and worked hard to figure out the shape of the sampling model. The model he found has become known as Student’s t. The Student’s t models are actually a whole family of models based on a parameter called the degrees of freedom that is determined by the sample size.

More on Student’s t As a result of correcting for the extra variation introduced from using the sample standard deviation in place of the population’s standard deviation: Confidence Intervals will be a little wider than with the Normal model P-values will be a little higher than with the Normal model Using the t-model is the correct way to deal with this additional variation

Confidence Interval When the conditions are met, we get: with the standard error given by: The critical value is determined by the confidence level we set and the degrees of freedom that comes from the sample size, n

Student’s t vs. Normal model Student’s t models are unimodal, symmetric and bell-shaped like Normal model But for small sample sizes, the Student’s t model has much fatter tails than the Normal As sample size increases, Student’s t model look more like the Normal model Figure from DeVeaux, Intro to Stats

Assumptions and Conditions Gosset found this model by simulation, but Sir Ronald A. Fisher later showed he was correct mathematically with some additional assumptions required. Independence Assumption Independence Randomization Condition 10% Condition

Example: Parking Garage Revenue During a two-month period (44 weekdays), daily fees collected averaged $126, with a standard deviation of $15. What assumptions do we need to make to do a statistical analysis? Find a 90% confidence interval for the mean daily income this parking garage will generate. Explain in context what this confidence interval means. Example from DeVeaux, Intro to Stats

Normal Population Assumption In addition to our Independence Assumption, we need to see that our data is nearly Normal Normal Population Assumption Nearly Normal Condition: distribution of data is unimodal and symmetric as verified with a histogram The smaller the sample size (under 15 or so) the more close to a Normal curve the data should be For larger (15 – 40 or so), t works well as long as the data are unimodal and symmetric For sample sizes over 40 or 50, t methods are safe to use unless data is very skewed

Hypothesis Test for Means Assumptions are same as for confidence interval We test H0:  = 0 using: again, with standard error: When conditions are met and null hypothesis is true, t follows Student’s t model with n-1 degrees of freedom and use that model for P-value

Student’s t Table We use a table in our text to find the P-value Student’s t varies for different degrees of freedom Figure from DeVeaux, Intro to Stats

Example: Battery Life A company claims that its battery lasts at least 7 hours. The average battery use time for a sample of 100 laptop batteries is found to be 7.3 hours with a standard deviation of 1.9 hours. Test the company’s claim and calculate the P-value. State an appropriate conclusion.

Determining Sample Size If we know our Margin of Error (ME) we can determine the sample size for our confidence interval as before. But now we won’t know tn-1 or s We can use s from a small pilot study We can use z instead of t

Minitab Example: Yogurt Consumer Reports tested 14 brands of vanilla yogurt and found the following numbers of calories per serving: 160 200 220 230 120 180 140 130 170 190 80 120 100 170 Check if the assumptions and conditions for inference are met. Create a 95% confidence interval for the average calories content of vanilla yogurt. A diet guide claims that you will get 120 calories from a serving of vanilla yogurt. Does your confidence interval support that claim? Example from DeVeaux, Intro to Stats