How many colleges did you apply to?

How many colleges did you apply to?

Type the number into your clicker and hit “send”

10-2Estimating a Population Mean(σ Unknown)

Confidence Intervals in the Calculator

• High School students who take the SAT Mathematics exam a second time generally score higher than on their first try. The change in the score has a Normal distribution with standard deviation σ=50. A random sample of 250 students gain on average x-bar=22 points on their second try.

• Construct a 95% Confidence interval for μ

Confidence Intervals Involving ZUsing the Calculator

What if we don’t know In common practice, we would never know the

population standard deviation. • Instead, we would use an estimate of : the

sample standard deviation, s. • We then estimate the standard deviation of

using • This is called the standard error of the sample

mean

“Standard error”: You are estimating the standard deviation…but there will likely be

some error involved because we are estimating it from sample data.

In other words… the standard error is (most likely) an inaccurate estimate of a (population)

standard deviation.

The t distributions

When we substitute the standard error of ()for its standard deviation () we get the distribution of

the resulting statistic, t.

We call it the t distribution.

The t-statistic was introduced in 1908 by William Sealy Gosset, a chemist working for the Guinness brewery in Dublin, Ireland ("Student" was his pen

name). Gosset devised the t-test as a way to cheaply monitor the quality of stout.

The t distributionsThere is a different t-distribution for each sample

size n.

We specify a t distribution by giving its degrees of freedom, which is equal to n-1

We will write the t distribution with k degrees of freedom as t(k) for short.

We also will refer to the standard Normal distribution as the z-distribution.

Comparing t and z distributions

Compare the shape, center, and spread of the t-distribution with

the z-distribution.

As the degrees of freedom k increase, (the sample size increases), the t-distribution is

increasingly Normal.

Our formula is the same as it was for z-intervals EXCEPT we replace sigma with s!!!

Finding t with Table CSuppose you

want to construct a 95%

confidence interval for the mean μ of a

population based on a SRS of size

n=12. What critical value t

should you use?

Finding t with Table CSuppose you want to construct a 95% confidence interval for the mean μ of a

population based on a SRS of size n=12. What critical value t should you use?

Finding t with Table C

Suppose you want to construct a 90% confidence interval for the mean μ of a population based on a

SRS of size n=15. What critical value t should you use?

Finding t with Table CSuppose you want to construct a 99% confidence interval for the mean μ of a population based on a SRS of size n=34. What critical value t should you

use?

Suppose you want to construct a 80% confidence interval for the mean μ of a population based on a

SRS of size n=95. What critical value t should you use?

a) 1.290b) .846c) 1.292c) .845

One sample t interval for 1)SRS2) Normality

- n < 15 : Use t procedures if data are close to Normal with no outliers

- n ≥ 15 : Use t procedures except in cases of outliers of strong skew

- n ≥ 30 : Use t-procedures even for clearly skewed distributions (cannot have extreme

outliers)3) Independence

One sample t interval for

Let’s use our class data to construct a 95% confidence interval for the true mean number of colleges that high school seniors applied to in

2013.

One sample t interval for mu

Step 1: STATEStep 2: PLAN

Step 3: CALCULATIONSStep 4: INTERPERATION

State: We are estimating ________, the true mean

______________________________________________________________.

Plan:Procedure:Conditions: 1)

2)

3)

Calculations:

Interpretation: We are 95% confident that the true mean

“Last year, 750,000 applicants submitted 3 million applications, an average of four per student”

College Decision Day: More Applications, More Problems|TIME.com

 http://nation.time.com/2013/05/01/as-college-applications-rise-so-does-indecision/#ixzz2sr0ANbp4

Which of the following changes will make a t-distribution more Normal?

a) Decrease b) Increase the Confidence Levelc) Decrease the margin of error. d) Increase

Paired t-proceduresTo compare the responses of the two treatments in a

matched pairs design or before and after measurements on the same subjects, apply the one sample t procedures to the differences observed between the pairs.

• µ = the mean difference between each pair

Ex) Mrs. Skaff gave a new study tool to her students to see if it would improve their test scores. She matched students based on current grade and randomly gave one student in each pair the study tool.

Paired t-procedures• µ = the mean difference between each pair Ex) Mrs. Skaff gave a new study tool to her students to see if it would improve their test

scores. She matched students based on current grade and randomly gave one student in each pair the study tool. She wants to know if the study tool improved test scores.

92 73 81 89 95 90 96 72 85 8890 73 84 84 88 91 93 70 80 882 0 -3 5 7 -1 3 2 5 0

Given Study ToolNo Study ToolStudy– No Study

Confidence Intervals in the Calculator

You still need all other steps!!!!

Ronald McDonald’s sister Diana Rhea is the purchasing manager for domestic hamburger

outlets. The company has decided to provide a free package of Tums to any complaining

customer. In order to estimate monthly demand, she took a sample of 5 outlets and found the number of Tums distributed to customers in a

month was250, 280, 220, 280, 320

(a)Find the sample mean and sample standard deviation

(b)Construct a 90% confidence interval on the average monthly demand per outlet.

Homework!