ch 8. confidence intervals part ii

CONFIDENCE INTERVALSCHAPTER 8 – PART II

THE STUDENT t DISTRIBUTION

The Student t distribution is a continuous distribution

The graph of the Student t distribution is bell shaped (so unimodal and symmetric)

Confidence Interval for a Mean Student t Distribution Confidence Interval for a Proportion


The mean, median, and mode of the Student t distribution are all equal to 0 and located at the peak

The Student t distribution is symmetrical about its mean. Half the area under the curve is above the peak, and the other half is below it



The Student t distribution is very similar to the standard normal distribution The Student t distribution has mean equal to 0, and depends upon the degrees of freedom

df

The That is, the degrees of freedom are one less than the sample size

We will often just call it a t distribution Just like in the case of the normal distribution, we will have to use a t-table to calculate

probabilities


CONNECTION TO THE STANDARD NORMAL DISTRIBUTION

The Student t distribution is very similar to the standard normal distribution

The standard normal distribution is a t distribution when the


CONFIDENCE INTERVAL FOR THE MEAN

When the population standard deviation is unknown, the CL% confidence interval is

So when the population standard deviation is unknown,

Important: is NOT the degrees of freedom is called the critical value s is the sample standard deviation which we use since the population standard deviation is

unknown In most practical applications, we usually do not know the population standard deviation

and have to use the sample standard deviation


CALCULATING THE CRITICAL VALUES

Let . Find the critical value when the sample size is 10. Solution: Using your t-distribution chart, find the value for when:

n = 10 df = n -1 = 9

2.262



Let . Find the critical value when the sample size is 8. Solution: Using your t-distribution chart, find the value for when:

n = 8 df = n -1 = 7

1.895



Let . Find the critical value when the sample size is 1000. Solution: Using your calculator, find the value for when:

n = 1000 Under DISTR, use invT(

invT(area, df) df = n -1 = 999

1.645


CALCULATING CONFIDENCE INTERVALS FOR THE MEAN WITH UNKNOWN Ç The speeds of 20 vehicles are

observed by radar on a particular road. For the vehicles in the sample, the average speed is 31.3 miles per hour with a standard deviation of 7.0 mph. Construct and interpret a 98%

confidence interval estimate of the true population average speed of all vehicles traveling on this road. Use a 98% confidence level.

s = 7.0 n = 20 df = n-1 = 19

We are 98% confident that the true population average speed is between 27.326 mph and 35.274 mph.


EXAMPLE 4 – YOU TRY IT!

In a random sample of 13 American adults, the mean waste recycled per person per day was 5.3 pounds and the standard deviation was 2.0 pounds. Construct a 95% confidence interval for the population mean.

s = 2.0 n = 13 df = n-1 = 12

179

Thus, we are 95% confident that the true population average waste recycled per person per day is between 4.091 pounds and 6.509 pounds.


ch 8. confidence intervals part ii

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