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How many colleges did you apply to?. Type the number into your clicker and hit “send”. 10-2 Estimating a Population Mean (σ Unknown). Confidence Intervals in the Calculator.

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How many colleges did you apply to

How many colleges did you apply to?

Type the number into your clicker and hit “send”


10 2 estimating a population mean unknown

10-2

Estimating a Population Mean

(σ Unknown)


Confidence intervals in the calculator

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 z using the calculator

Confidence Intervals Involving Z

Using the Calculator


What if we don t know

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


How many colleges did you apply to

“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 distributions


How many colleges did you apply to

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.


There is a different t distribution for each sample size n

There is a differentt-distribution for each sample size n.

The t distributions

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

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.


How many colleges did you apply to

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


Finding t with table c

Suppose 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


Finding t with table c1

Suppose 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


Finding t with table c2

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 C


Finding t with table c3

Suppose 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?

Finding t with Table C


How many colleges did you apply to

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) 1.290

b) .846

c) 1.292

c) .845


One sample t interval for

One sample t interval for

1)SRS

2) 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 for1

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


Step 1 state

Step 1: STATE

One sample t interval for mu

Step 2: PLAN

Step 3: CALCULATIONS

Step 4: INTERPERATION


How many colleges did you apply to

  • State: We are estimating ________, the true mean ________________________________

  • ______________________________.

  • Plan:

  • Procedure:

  • Conditions: 1)

  • 2)

  • 3)


How many colleges did you apply to

  • Calculations:

  • Interpretation: We are 95% confident that the true mean


How many colleges did you apply to

  • “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

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

  • Decrease

  • Increase the Confidence Level

  • Decrease the margin of error.

  • Increase


Paired t procedures

Paired t-procedures

  • To 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 differencesobserved 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 procedures1

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.

Given Study Tool

No Study Tool

Study– No Study


Confidence intervals in the calculator you still need all other steps

Confidence Intervals in the CalculatorYou still need all other steps!!!!


How many colleges did you apply to

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 was

250, 280, 220, 280, 320

Find the sample mean and sample standard deviation

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


Homework

Homework!


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