T Distribution

  • When do we use the T distribution

    • Know the mean

    • Do not know the population SD

    Z distribution (standard normal) t-distribution (n close to 30)
t-distriöution (n smaller than 30)

  • Formula

    Image result for confidence interval t

  • 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

    "-84 Pius Silver Ecfitjm TEXAS INSTRUMENTS FORMAT n

  • 1-Var Stats for the sample

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  • Looking for t

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  • Calculate

    ー ひ , ツ 亠 )

  • Calculate by calculator

    Tl-B4 Pius Silver Ecfitjm TEXAS INSTRUMENTS FORMAT n FS

  • 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

    Test stat (Point Estimate) — (Null Hypothesis) Standard Error

    -2.29 3/450

  • Calculate by calculator

    Tl-B4 Pius Silver Ecfitjm TEXAS INSTRUMENTS FORMAT n

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

  • P-value

    QI

  • 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?

Image result for matched pairs t
 test

  • Formula

    C:\\6432CA65\\FE01530B-89BD-4F8B-A3E1-55F12080AD12\_files\\image247.png

  • Data

    • N=10

    • Difference of mean = 0.52

    • Difference of SD = 0.407

  • Hypothesis (d = before - after)

    • H0: µd = 0

    • H1: µd > 0

  • Conditions

    • Random: assume random selection

    • Independence: N > 10n = 100

    • Normal: t-distribution

  • Calculate

    01 —

  • Calculate by calculator

    Tl-B4 Pius Silver Editjm TEXAS INSTRUMENTS FORMAT n

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

  • P-value

    • df = 10-1 = 9

    C:\\6432CA65\\FE01530B-89BD-4F8B-A3E1-55F12080AD12\_files\\image251.png

  • Interpret

    • P < α, Reject, do have evidence

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