Inference for Means The t-distribution AP Statistics

Remember this? Suppose we’re testing a new antibacterial cream, “Formulation NS,” on a small cut made on the inner forearm. We know from previous research that with no medication, the mean healing time (defined as the time for the scab to fall off) is 7.6 days, with a standard deviation of 1.4 days. The claim we want to test here is that Formulation NS speeds healing. We will use a 5% significance level. We cut 25 volunteer college students and apply Formulation NS to the wound. The mean healing time for the subjects is 7.1 days. We will assume that we know the population standard deviation, and that it’s 1.4 days.

When might we actually know s? Administrators at a hospital were concerned about the prenatal care given to mothers in their part of the city. To study this, they examined the gestation times of babies born there. The drew a sample of 25 babies born in their hospital in the previous 6 months. Human gestation times for healthy pregnancies are known to be well modeled by a Normal model with a mean of 266 days and a standard deviation of 16 days.

I’m taking s away!!

William S. Gosset

t – test statistic Draw an SRS of size n from a population that has the normal distribution with mean m and standard deviation s. The one sample t statistic has the t distribution with n-1 degrees of freedom.

Ok, so let’s look at a problem. Cola makers test new recipes for loss of sweetness during storage. Trained tasters rate the sweetness before and after storage. Here are the sweetness losses (sweetness before storage minus sweetness after storage) found by 10 tasters for one new cola recipe: 2.0 0.4 0.7 2.0 -0.4 2.2 -1.3 1.2 1.1 2.3 Are these data good evidence that the cola lost sweetness?

= the mean loss in sweetness Parameter: = the mean loss in sweetness Hypotheses: H0: m = 0 Ha: m > 0

Assumptions & Conditions Our data is a simple random sample (SRS) of size n from the population of interest. This condition is very important Observations from the population have a normal distribution with mean m and standard deviation s. Check for symmetry and outliers

SRS: We must be willing to treat the 10 tasters as an SRS SRS: We must be willing to treat the 10 tasters as an SRS. Normal: This doesn’t look good, but there are no outliers, so we’ll proceed with caution.

Name: We’ll run a one-sample t-test Name: We’ll run a one-sample t-test. Test statistic: df = n – 1 = 10 – 1 = 9

Obtain P-Value: P-value = P(t > 2.70) between .02 and .01 By calc: 0.012 Make Decision: This low P-value suggests We should reject H0. State in context: The low probability provides evidence to suggest that the cola has lost sweetness during storage.

Computer Output

How about confidence intervals? Environmentalists, government officials, and vehicle manufacturers are all interested in studying the auto exhaust emissions produced by motor vehicles. The major pollutants in auto exhaust are hydrocarbons, monoxide, and nitrogen oxide (NOX). The NOX levels (in grams/mile) for a sample of light-duty engines of the same type are listed on the next slide. Construct a 95% confidence interval for the mean amount of NOX emitted by light-duty engines of this type.

Sample Data 95% Confidence Interval: 1.28 1.17 1.16 1.08 1.32 1.24 0.71 0.49 1.38 1.20 0.78 0.95 2.20 1.78 1.83 1.26 1.73 1.31 1.80 1.15 0.97 1.12 0.72 1.31 1.45 1.22 1.32 1.47 1.44 0.51 1.49 1.33 0.86 0.57 1.79 2.27 1.87 2.94 1.16 1.45 1.51 1.47 1.06 2.01 1.39 1.60 95% Confidence Interval:

(1.185, 1.473)