Chapter 10 Section 1.pptx

Chapter 10 Section 1

Statistical inference is the process by which we acquire information and draw conclusions about populations from samples. In order to do inference,

we require the skills and knowledge of descriptive statistics, probability distributions, and sampling

Estimating with Confidence

Suppose I want to know how often teenagers go to the movies. Specifically, I want to know how many times per month a typical teenager (ages 13

through 17) goes to the movies.

Suppose I take an SRS of 100 teenagers and calculate the sample mean to be .

The sample mean is an unbiased estimator of the

unknown population mean _____, so I would estimate the

population mean to be approximately 2.1. However, a

different sample would have given a different sample

mean, so I must consider the amount of variation in the sampling

model for _____.

The sampling model for is approximately normal. The mean of the sampling model is _____.

The standard deviation of the sampling model is ______ assuming the population size is at least 30.

m

x

x

Chapter 10 Section 1

Suppose we know that the population standard deviation

is . Then the standard deviation for the sampling model is .

Then 95% of our samples will produce a sample mean

that is between 2 and 2.2.

Therefore in 95% of our samples, the interval between

2 and 2.2will contain the parameter .

The margin of error is 0.10. s

n =

0.5

100 =.05

s =

0.5

x

For our sample of 100 teenagers, . Because the margin of error is 0.10, then we are 95% confident that the true population mean lies somewhere in the interval [ 2 , 2.2 ], or [2.1 .10].  

The interval [2.0, 2.2] is a 95% confidence interval

because we are 95% confident that the unknown lies between 2.0

and 2.2.

Start with sample data. Compute an interval that has probability C of containing the true value of the parameter. This is called a

Level C confidence interval.

x

=

2.5

How do we construct confidence intervals?

Since the sampling model of the sample mean is

approximately normal, we can use normal

For a 99% confidence interval, we want the interval

corresponding to the middle 99% of the normal curve. Z-score for 99%: 2.576

For a 95% confidence interval, we want the interval

corresponding to the middle 95% of the normal curve

Z-score for 95%: 1.96

For a 90% confidence interval, we want the interval

corresponding to the middle 90% of the normal curve. Z-score for 90%: 1.645

Confidence Interval for a Population Mean

If we are using the standard normal curve, we want to

find the interval using z-values.

Suppose we want to find a 90% confidence interval for a standard normal curve. If the middle 90% lies within our interval, then the remaining 10% lies outside our interval. Because the curve is symmetric, there is 5% below the interval and 5% above the interval.

Find the z-score value with area 5% below and 5% above. -1.645 & 1.645

These z-values are denoted Z*. Because they

come from the standard normal curve, they are centered

at mean 0.

Z* is called the upper p critical value, with

probability p lying to its right under the standard normal

To find p, we find the complement of C and divide it in

half, or find 1-C

For a 95% confidence interval, we want the z-values with

upper p critical value 2.5%.

For a 99% confidence interval, we want the z-values with

upper p critical value 0.5%.

Remember that z-values tell us how many standard deviations

we are above or

below the mean.

To construct a 95% confidence interval, we want to find

the values 1.96 standard deviations below the mean and

Page 544

Using our sample data, this is ,

assuming the population is at least 10n.

In general, to construct a level C confidence interval

using our sample data, we want to find

.

The margin of error is __________. Note that the margin

of error is a positive number. It is not an interval.

x

±

1.96

s

n

x

±

z

*

s

n

Confidence Interval for a Population Mean (pg. 546)

Draw an SRS of size n from a population having unknown

mean and known standard deviation . A level C confidence

interval for is

Here z* is the value with area C between –z* and z* under the standard normal

curve. This interval is exact when the population distribution is normal and is

approximately correct for large n in other cases.

m

_s

Inference Toolbox ~ Confidence Intervals

To construct a confidence interval: PANIC

P-

A-

N-

I-

C-

Chapter 10 Section 1

Conclusion

Ex: We are _____ % confident that the true

population mean, , lies between ____ and ____.

Example #1: Suppose that the population standard deviation for the weight of all male Americans is 15 pounds. Find a 95% confidence interval for the mean weight of all male Americans based on a simple

How Confidence Intervals Behave

We would like high confidence and a small margin of error.

A higher confidence level means a higher percentage

of all samples produce a statistic close to the true

value of the parameter. Therefore we want a

High level of confidence.

A smaller margin of error allows us to get closer to the true

value of the parameter, so we want a low margin of

error.

So how do we reduce the margin of error?

Lower the confidence level (by decreasing the value

of z*).

Increase the standard deviation.

Increase the sample size. To cut the margin of error

in half, increase the sample size by FOUR times

You can have high confidence and a low

margin of error if you choose the right sample size.

To determine the sample size n that will yield a confidence interval for a population mean with a specified margin of error m, set the expression for the margin of error to be less than or equal to m and solve for n.

Sample size for desired margin of error:

z

*

s

Example #2: A lumber company must estimate the mean diameter of trees to determine whether or not there is sufficient lumber to harvest an area of forest. They need to estimate this to within 1 inch at a

confidence level of 99%. The tree diameters are normally distributed with a standard deviation of 6 inches. How many trees need to be sampled?

CAUTIONS!! (pg. 553-554)

These methods only apply to certain situations. In

order to construct a level C confidence interval using the

formula , for example, the data must be an

SRS and we must know the population standard deviation.

Also, we want to eliminate (if possible) any outliers.

The margin of error only covers random sampling errors.

Things like undercoverage, nonresponse, and poor

sampling designs can cause additional errors.

Advanced Placement Statistics

Confidence Intervals (Population mean) Worksheet

A study of forty dogs showed that they could bark for an average of 15 minutes.  The standard deviation is 0.6.  Find the 99% confidence interval for the mean of all dogs.

A researcher has studied fifty people and found that the average number of hours a person watches television is three hours a day.  The standard deviation is 2.  Find the 95% confidence interval for the mean of all people.

A random sample of thirty-six animals is taken at a veterinarian clinic.  The veterinarian finds that the average number of sick animals is 15.  The standard deviation is 3.  Find the 90% confidence interval for the mean of all animals.

A random sample of thirty-one customers shows

that the average customer spends thirty dollars per visit.  The standard deviation is 5.  Find the 99%

A study conducted of fifty lumberjacks showed that they could cut down an average of six trees an hour.  The

standard deviation is 1.  Find the 95% confidence interval for the mean of the lumberjacks.

A researcher studied 100 people and found that the average number of pets per household is 1.5.  The standard deviation is 0.3.  Find the 80% confidence interval for the mean number of pets per household.

A random sample of 150 people shows that the average number of cars per household is 2.3.  The standard

deviation is 0.4.  Find the 90% confidence interval for the mean number of cars per household.

A study conducted of 75 technicians showed that they

could average 3 technical problems an hour. The standard deviation is 0.2. Find the 95% confidence interval for the mean number of technical problems per hour.