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?
Type the number into your clicker and hit “send”
10-2
Estimating a Population Mean
(σ Unknown)
Confidence Intervals Involving Z
Using the Calculator
“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.
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.
There is a differentt-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.
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!!!
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?
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?
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?
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?
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
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
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.
Step 1: STATE
Step 2: PLAN
Step 3: CALCULATIONS
Step 4: INTERPERATION
Given Study Tool
No Study Tool
Study– No Study
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.