Confidence Intervals: estimate z* (standard dev of estimate)

z* from Normal Probability Table

e.g. Means z* ; Proportions: z*

Tests of Significance: Assume H_o is true, and calculate a test statistic to see if observation is very rare (compare distance between estimate and hypothesized value dividing by std dev of estimate).

Means: z₀=(- m₀)/(); Proportions: z_o = ( - p₀)/

• One sample (includes matched pairs design)

• Data from an SRS
• Normality of sampling distribution of statistic

• When dealing with , know value of s

Standard deviation()=

Standard Error of a Statistic:

When we don’t know the standard deviation exactly, we can estimate it from the data.

1. Std Dev() = , substitute , SE()=

2. Std Dev() =, substitute s, SE()=

"standard error of " (Explains SE MEAN in Minitab output).

The bad news: With means, longer follows a Standard Normal Distribution!

What should be true about the distribution it does follow?

In fact, ( - m)/() has a Student’s t distribution with n-1 degrees of freedom, t_n-1 IF

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Properties of Student’s t distribution:

Calculating p-values from Student's t distribution:

Table D gives you some information about p-value.

 

Example Is there significant evidence that the gas mileage increase is greater than zero?

 

 

Calculating critical values from Student’s t distribution:

For 95% confidence interval, find t* (Table D)

How does t* compare to z*(95%)?

Why does this make sense?

Example 95% confidence interval for mileage difference:

 

 

SUMMARY Inference when s is unknown:

Hypothesis Test:

Confidence Interval:

Use Table D for critical values and p-values.

Technical assumptions of Student t procedures for means:

• Simple Random Sample (See how the data was collected)
• Normality of

1. Normality of measurements

Can be hard to check

- past experience (small data sets)

- plots (Eliminate outliers?) (Normal Probability Plot)

or 2. Large n

Use 30/40 for cutoff between large and small sample inference

Usually safe with t procedures when n > 15 (robust)

If n is small check for skewness and outliers.