statistics class 16

Statistics Class 16

3/26/2012

Estimating a Population mean: known

We are going to use the point estimate , to estimate the population mean using confidence intervals, when is known.



• Usually you don’t know if you don’t know the population mean.



• Usually you don’t know if you don’t know the population mean.

• We require either normality or for the most part. We actually require a loose normality that is that there are not too many outliers and the data sort of has a bell shape.


Constructing a Confidence Interval for (with known)1. Verify that the requirements are satisfied.


Constructing a Confidence Interval for (with known)1. Verify that the requirements are satisfied. (Sample is

simple random, is known, and distribution is either normal or

2. Find the critical value associated with the desired confidence level.





3. Evaluate the margin of error





3. Evaluate the margin of error 4. Construct


Constructing a Confidence Interval for (with known)5. Round

a. If using the original data set, round to one more decimal than the original set

b. If using summary statistics round to the same place as the sample mean.

OrZinterval


A simple random sample of 40 salaries of NCAA football coaches has a mean of $415,953. Assume = $463,364.• Find the best point estimate of the mean salary of all NCAA

football coaches.

• Construct a 95% confidence interval estimate of the mean salary of NCAA football coach.

• Does the confidence interval contain the actual population of mean of $474,477?


Determining the sample size Required to Estimate The sample size required to estimate is given by the formula:

Round up to the next larger whole number.


How many integrated circuits must be randomly selected and tested for time to failure in order to estimate the mean time to failure? We want a 95% confidence that the sample mean is within 2 hr. of the population mean, and the population standard deviation is known to be 18.6 hours.

Estimating a Population mean: Not known

In this section we learn to estimate the population mean when is not known. Since is typically unknown in real life this is a useful method as it is practical and realistic.

The sample mean is the best point estimate of the population mean .

When is not know we use a Student t distribution to find our critical values.


If a population is normally distributed, then the distribution of

is a Student t distribution for all samples of size n.


If a population is normally distributed, then the distribution of

is a Student t distribution for all samples of size n.

The number of degrees of freedom for a collection of sample data is the number of sample values that can vary after certain restrictions have been imposed on all data values. The number of degrees of freedom is often abbreviated as df.


For us the number of degrees of freedom is simply the sample size minus 1.

Degrees of freedom


For us the number of degrees of freedom is simply the sample size minus 1.

Degrees of freedom

Find the critical t value corresponding to a confidence level of 95% for a sample of size .


Constructing a Confidence Interval for (with unknown)1. Verify requirements are satisfied(Simple random sample,

normal dist. or ).



normal dist. or ).2. Using degrees of freedom find the critical value

corresponding to the desired confidence level.




corresponding to the desired confidence level.3. Evaluate the margin of error .




corresponding to the desired confidence level.3. Evaluate the margin of error .4. Find .




corresponding to the desired confidence level.3. Evaluate the margin of error .4. Find .5. Round to one extra decimal place if using original data. If

using summary statistics round to the same number of decimal places as the sample mean.


Or use Tinterval


In a test of the Atkins weight loss program, 40 individuals participated in a randomized trial with overweight adults. After 12 months, the mean weight loss was found to be 2.1 lb, with a standard deviation of 4.8 lb• What is the best point estimate of the mean weight loss of all

overweight adults who follow the Atkins program?• Construct a 99% confidence interval estimate of the mean

weight loss for all such subjects.• Does the Atkins program appear to be effective? Is it

practical?


Important properties of the t distribution.1. The Student t distribution is different for different sample

sizes.



sizes.2. The Student t distribution has the same general

symmetric bell shape as the standard normal distribution, but reflects the greater variability that is expected with small samples.





3. The Student t distribution has a mean of t=0





3. The Student t distribution has a mean of t=04. The standard deviation of a t distribution varies with the

sample size but is always larger than 1.



sizes.2. The Student t distribution has the same general symmetric

bell shape as the standard normal distribution, but reflects the greater variability that is expected with small samples.

3. The Student t distribution has a mean of t=04. The standard deviation of a t distribution varies with the

sample size but is always larger than 1.5. As n gets large the t distribution becomes more like the

standard normal distribution.


Method Conditions

Use z distribution known and normal dist. Pop.or and n > 30

Use t distribution unknown and normal dist. Pop.or unknown and n > 30

Other method If n and Population not normal


Use the given data to decide whether to use ZInterval or Tinterval• =75, s=15 and population has a normal dist.• and the population is very skewed.• and the population is normal.• and the population is very skewed• and the population is very skewed.


Listed below are 12 lengths (in minutes) of randomly selected movies. 110 96 125 94 132 120 136 154 149 94 119 132• Construct a 99% confidence interval estimate of the mean

length of all movies.• Assuming that it takes 30 min to empty a theater after a

movie, clean it, allow time for the next audience to enter, and show previews, what is the minimum time that a theater manager should plan between start times of movies, assuming that this time will be sufficient for typical movies?

Quiz 13

When Mendel conducted his famous genetics experiments with peas, one sample of offspring consisted of 428 green peas and 152 yellow peas.a. Find a 95% confidence interval estimate of the percentage

of yellow peas.b. Based on his theory of genetics, Mendel expected that

25% of the offspring peas would be yellow. Given that the percentage of offspring yellow peas is not 25%, do the results contradict Mendel’s Theory? Why or why not.

Homework!!

• 7-3: 1-9 odd, 13,15, 21-27 odd.• 7-4: 1-12, 17-29 odd.

statistics class 16

Documents

student t distribution

mean of t

standard normal distribution

sample mean

different sample sizes

mean time

mean weight loss

mean length