Estimating and Constructing Confidence Intervals

10.1 Interval EstimationFinding a value for or

95%

99%

2ZZ

96.12Z

576.22Z

10.2 Confidence Intervals for the Mean with Known Population VarianceThe population mean is an unknown

parameter

is a statistic which estimates

We call a point estimate because its value is a point on the real number line

Unfortunately, if we sample from a continuous distribution,

x

x

0)( xP

Statisticians prefer interval estimates

E (the Error Tolerance) depends on the sample size, how certain we want to be that we are correct (Level of Confidence), and the amount of variability in the data

When is known has approximately a

standard normal

distribution.

Ex

n

xZ

Therefore,

1

22zZzP

1

22 nzx

nzxP

n

Zn

ZE

2

Notice that increasing the level of confidence, decreases the probability of error, however it also increases E creating a wider interval

 Notice as sample size increases, E decreases

creating a more narrow interval Notice the more variability in the population,

the larger E creating a wider interval 

ExampleA sample of 100 visa accounts were studied for the

amount of unpaid balance. 

Construct and interpret a 95% confidence intervalWe are 95% confident that the mean unpaid balance of visa accounts is between $619.13 and $670.87.

Construct a 99% confidence intervalWe are 99% confident that the mean unpaid balance of visa accounts is between $611.00 and $679.00.

645$x132$

Choosing the Sample SizeIn the design stages of statistical research, it is

good to decide in advance the confidence level you wish to use and to select the error tolerance you want for the project. This will let us decide how big our sample needs to be.

Sample size for estimating

If the value of is not known, we do preliminary sampling to approximate it.

2

E

Zn

ExampleWe wish to estimate the number of patient-visit

hours per week physicians in solo practice spent. How large a sample is needed if we want to be 99% confident that our point estimate is within 1 hour of the population mean? Assume a standard deviation of 11.97 hours.

10.3 Student’s t Distribution is unknown

To avoid the error involved in replacing by , we will introduce a new random variable called Student’s t variable. (t-distribution)

If we sample from a normal distribution has a t-distribution with n-1 degrees of freedom.

s

nsx

t

Properties of the t-distributioncontinuous and symmetric about 0more variable and slightly different shape

than the standard normalAs n becomes large, the t distribution can be

approximated by the standard normal distribution (The bottom row of the t-distribution is Z)

With a sample size of 11 a Confidence level of 95%, what is the two tailed t value?228.210,025. t

10.4 Confidence Intervals for the Mean with Unknown Population Variance

For unknown, )(n

stE

ExampleMileage of tires in 1000’s of miles Sample: 42, 36, 46, 43, 41, 35, 43, 45, 40, 39

Compute a 95% confidence interval for

We are 95% confident that the population mean mileage of tires is between 38,432 and 43,568 miles.

10n

41x59.3s

262.29,05. t

ExampleA random sample of 20 apples yields and

Find a 99% confidence interval.

We are 99% confident that the population mean weight of apples is between 8.496 and 9.904 oz.

oz. 2.9x oz. 1.1s

10.5 Confidence Intervals for Proportions (Large Samples)

For large n, is approximately standard normal.

n

pq

ppZ

ˆ

n

qpZE

ˆˆ

Ep̂

ExampleA survey of 1,200 registered voters yields 540 who

plan to vote for the republican candidate. p = proportion of all voters who plan to vote for

the republican candidate

Find a 95% confidence interval for p

We are 95% confident that the population proportion of voters who plan to vote for the republican candidate is between 42.2% and 47.8%.