# T Distribution

When do we use the T distribution

Know the mean

Do not know the population SD

Formula

Degrees of freedom (df)

- df = n-1

# Confidence Interval Example

The super Corporation manufactures a device they claim "may increase gas mileage by 23%"

Here are the percent changes in gas mileage for 17 identical vehicles, as presented in one of the company's advertisements

48.3 | 46.9 | 46.8 | 44.6 | 40.2 | 38.5 |
---|---|---|---|---|---|

34.6 | 33.7 | 28.7 | 28.7 | 24.8 | 10.8 |

6.9 | 12.4 | 21.2 | 25.2 |

Construct and interpret a 90% confidence interval to estimate the mean fuel savings in the population of all such vehicles.

Type in the data

1-Var Stats for the sample

Looking for t

Calculate

Calculate by calculator

Interpret

We are 90% confident that the mean percent change in gas mileage is between 24.68% and 38.16%

Since 23% is in the interval, it appears that the machine does even better than 23% savings

# Hypothesis Test Example

When the manufacturing process is working properly, the batteries have lifetimes that follow a right-skewed distribution with µ = 10 hours. A quality control statistician selects a simple random sample of n = 20 batteries every hour and measures the lifetime of each.

If she is convinced that the mean lifetime of all batteries produced that hour is less than 10 hours at the 5% significance level, then all those batteries are discarded.

The current sample of 20 has a mean of 8.5 and a SD of 3, should this batch be discarded?

Hypothesis

H0: µ = 10

H1: µ < 10

Conditions

Random: given

Independent: N > 10n = 200

Normal: not larger than 30, but still ok since we are using a T distribution

Calculate

Calculate by calculator

![Т1-И Pius Stver Editj(T ф TEns [NSTRUMENTS гоямдта сасса ](./media/image244.png)

P-value

df = n - 1 = 19

Interpret

- Because p < α, we reject the null hypothesis. Thus, we do have evidence to support the claim that the mean battery life in this whole batch is less than 10 hours and so should be discarded

# Matched Pairs T-Test

The average weekly loss of study hours due to consuming too much alcohol on the weekend is studied on 10 students before and after a certain alcohol aware ness program is put into operation. Do the data provide evidence that the program was effective?

Formula

Data

N=10

Difference of mean = 0.52

Difference of SD = 0.407

Hypothesis (d = before - after)

H

_{0}: µ_{d}= 0H

_{1}: µ_{d}> 0

Conditions

Random: assume random selection

Independence: N > 10n = 100

Normal: t-distribution

Calculate

Calculate by calculator

![ТИ-84 PIus Stver Editj(n ф TEns [NSTRUMENTS гоямдта сац:а ](./media/image250.png)

P-value

- df = 10-1 = 9

Interpret

- P < α, Reject, do have evidence