Estimating a Population Mean Textbook Section 8.3

Inference and a Problem

 When we don’t know μ or σ, we cannot use the Normal Distribution (z-scores).  We have to use a new distribution – the t distributions.  Notice the plural – there is a different t distribution for each sample size.  We specify a particular t distribution by giving its degrees of freedom (df).  Degrees of freedom is found by subtracting 1 from the sample size: df = n – 1 When σ is unknown: The t Distributions

 The density curve in t distributions are similar to a Normal curve – single peaked and symmetric.  The spread of t is a bit greater than that of the Standard Normal distribution.  As the degrees of freedom increase, the t distribution will approach the Standard Normal Curve.  Explore Table B  When the df you need is not listed, round down. T- Distributions

 Use table B to find the critical value t* that you would use for a confidence interval for a population mean μ in each of the following settings.  A 96% confidence interval based on a random sample of 22 observations.  A 99% confidence interval from an SRS of 71 observations. Check your Understanding

 The Random condition is crucial for ALL inference.  The 10% condition ensures independence of samples.  Our sampling distribution must also be approximately Normal, which can be checked one of two ways.  The population is KNOWN to be approximately Normal.  Our sample size is large enough (n ≥ 30) to apply The Central Limit Theorem.  What if our sample size is less than 30??  You must plot your data (on your calculator is fine if you then sketch it) – I recommend a Box Plot.  Use t distribution ONLY if your plot has no outliers nor strong skewness!  ALWAYS CHECK YOUR CONDITIONS – AND PROVE IT!! Conditions for Estimating μ

 To estimate the average GPA of students at your school, you randomly select 50 students from classes you take. A histogram of GPAs is shown.  How much force does it take to pull wood apart? A stemplot of the force required to pull apart a random sample of 20 pieces of Douglas Fir is shown.  Suppose you want to estimate the mean SAT Math score at a large high school. A boxplot of the SAT Math scores for a random sample of 20 students is shown. Can we use t?

Constructing a Confidence Interval for μ

 A manufacturer of high-resolution video terminals must control the tension on the mesh of fine wires that lies behind the surface of the viewing screen. Some variation is inherent in the production process. Here are the tension readings from a random sample of 20 screens from a single day’s production.  Construct and interpret a 90% confidence interval for the mean tension of all the screens produced on this day.  Begin by entering the data into a list. Example: Video Screen Tension

Required Steps

Using Calculator

Choosing the Sample Size