AP Statistics11.1 Inference for the Mean of a

Population

Perform a one sample t-test for the mean of a population

Perform a matched pairs t-test

Learning Objective:

Our data are a simple random sample (SRS) of size n from the population

Observations from the population have a normal distribution with mean (µ) and standard deviation (σ). Both µ and σ are unknown parameters.

Assumptions for Inference about a Mean

When the standard deviation of a statistic is estimated from the data, the result is called the standard error of the statistic. The standard error of the sample mean is :

s/√n

Standard Error

  Draw an SRS of size n from a population that has

a normal distribution with mean µ and standard deviation σ.

The one-sample t statistic 

has the t distribution with n-1 degrees of freedom.  

The one-sample t statistic and the t distributions

  The one sample t confidence interval has

the form

estimate ± t*SE =

Confidence Interval

(Refer to class activity)

How do degrees of freedom affect the shape of the t distribution?

The larger the df, the closer it gets to a normal dist.

How does the t distribution similar/different from the standard

normal curve?

One Sample t-test Matched Paired t-test

µ=true mean test score

H₀: µ=80Ha: µ>80

µ=true mean DIFFERENCE in test

scores

H₀: µ=0Ha: µ>0

Ex 1: Is it said that a person should get 8 hours of sleep per night. The editors of a school newspaper at Frontier High School wanted to know if this was true of the students at their school. A survey was conducted using a random sample of 16 students and each student was asked how many hours they slept the night before. Conduct a test to determine whether the students at Frontier high School were sleeping the recommended number of hours and state your conclusions.

6 5 7 4 4 8 7 66 4 9 5 7 6 5 8

µ=true mean number of hours slept each night H₀: µ=8 Ha: µ≠8 Assumptions-random sample-norm. distributed(w/ no extreme outliers)

One sample T-test w/ α=0.1df=15

Since p< α, it is statistically significant, therefore we reject H₀. There is enough evidence to say the true mean sleeping time is different than 8.

Give a 90% confidence interval for the mean number of hours students slept.

One sample t-interval

Assumptions same as abovedf=156.0625±0.6689=(5.3936, 6.7314)

We are 90% confident that the true mean numbers of hours slept is between 5.4 and 6.7 hours.

Ex 2:Cola makers test new recipes for loss of sweetness during storage. Trained testers 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. Are these data good evidence that the cola lost sweetness?

2.0 0.4 0.7 2.0 -0.4 2.2 -1.3 1.2 1.1 2.3

µ=true mean sweetness loss in cola H₀: µ=0 Ha: µ>0 Assumptions-random sample-norm. distributed(w/ no outliers)

Match Paired T-test w/ α=0.025 (df=9)

Since p< α, it is statistically significant, therefore we reject H₀. There is enough evidence to say the true mean sweetness loss is greater than 0.

Give a 95% confidence interval for the mean amount of sweetness loss in the cola.

Matched pairs t-interval Assumptions:-same as abovedf=91.0214 ± 0.8556 = (0.16436, 1.8756)We are 95% confident that the true mean

amount of sweetness loss is between 0.16 and 1.88

Ex 3: A random sample of 25 freshmen can do an average of 15 pushups with a standard deviation of 9 were taken after completing a special fitness education program. Is this value significantly greater than the average of 12 pushups for all freshmen at this school? Is there evidence that the physical education program increases the number of push-ups that can be done?

µ=true mean number of push ups H₀: µ=12 Ha: µ>12 Assumptions-random sample-norm. distributed(w/ no extreme outliers)

One sample T-test w/ α=0.025

(df=24)

Since p∡α, it is not statistically significant, therefore we do not reject H₀. There is not enough evidence to say the true mean number of push ups is greater than 12.

Ex 4: 11 students in a mathematics classroom had attended a review session for an upcoming test. The students took a test on the material before the review and after the review. Below are the students’ scores. Is there evidence that the review session significantly help the students’ test scores?

Subject Pre-test Post-test1 84 902 70 813 87 894 77 825 60 756 93 957 91 968 78 849 66 7910 63 7111 81 88

µ=true mean difference in the test scored H₀: µ=0 Ha: µ>0 Assumptions insert stemplot!!-random sample-norm. distributed(w/ no extreme outliers)

Match Pairs T-test w/ α=0.025df=10

Since p<α, it is statistically significant, therefore we do reject H₀. There is enough evidence to say the review session helped.