Deliverable 3 - Confidence Intervals

Module 03 – Confidence Intervals

Class Objectives:

· Introduce and define the t Distribution.

· Define population parameters and point estimates.

· Define and calculate a confidence interval.

· Determine the sample size needed to achieve a certain margin of error.

Module 03 - Part 1

In module 02, we introduced the normal distribution:

Recall that ________ = normal variable and ___________ = standard normal variable.

· The _________________________________ is a family of distributions that look almost identical to the normal distribution curve, only a bit fatter and shorter.

· The layout of the distribution depends on the _______________________________ ().

The t distribution can be used to calculate probabilities for ______________________________________ or when the population standard deviation __________________________________________________.

Just like for the normal distribution, we have functions in Excel to help us calculate the probability of a t-statistic and vice versa.

· How to find the probability of a t-statistic: _______________________________________.

· Note: this is only for a left-tailed value!

· How to find the t-statistic given a probability: _________________________________________.

· Note: this is only for a left-tailed value!

· A _________________________________________________ is any summary number, like an average or percentage, which describes the entire population.

· For example, one population parameter is _________ which represents the population mean.

The problem is that most of the time we don’t know the real value of the population parameter (unless we have a census). The best we can do is estimate!

· A ___________________________________ of a population parameter is a single value of a statistic.

· For example, the sample mean _____________ is the best point estimate of the population mean .

· ____________________________________________________ give us a range of values that we can be sure that a population parameter falls into a certain range given a point estimate.

· In other words, confidence intervals give us a range of values we are fairly sure our true value lies in.

· Why do we need confidence intervals?

A confidence interval consists of three parts:

1)

2)

3)

Example.

We measure the heights of 40 randomly chosen men, and get a mean height of 175cm. We also know the standard deviation of men’s heights is 20 cm.

The 95% confidence interval is 175cm 6.2 cm

This says the true mean of ALL men (if we could measure all their heights) is likely to be between 168.8cm and 181.2cm.

Caution: it might not be! The "95%" says that 95% of experiments like we just did will include the true mean, but 5% won't.

· The _________________________________ describes the likelihood that a particular sampling method will produce a confidence interval that includes the true population parameter.

· The most common confidence levels are ____________________________________________.

· In a confidence interval, the range of values above and below the point estimate is called the _______________________________________.

For our problems, we are going to concentrate on estimating the population mean () which means we need to use our sample mean () to create our confidence interval.

How to calculate the confidence interval for

1) Find the Critical Value ()

2) Calculate the Margin of Error ()

3) Add and Subtract the margin of error from the mean ,

4) Interpret the confidence interval

· “There is a 95% chance that the confidence interval *insert confidence interval* contains the true population mean.”

How to find the critical value

1) Calculate the degrees of freedom: ________________________.

2) Compute alpha: _____________________________________________.

3) Use the excel function: _____________________________________.

Examples.

a) Find the critical value corresponding to a 95% confidence level, given that the sample has size .

b) Find the critical value corresponding to a 99% confidence level, given that the sample has size .

Depending if your population standard deviation is known or not, you will either have to use a z or t score to calculate your margin of error.

· When is unknown, we use the following method to calculate the margin of error:

· Use degrees of freedom = ______________________

Example. Nine hundred (900) high school freshmen were randomly selected for a national survey. Among survey participants, the mean grade-point average (GPA) was 2.7, and the standard deviation was 0.4. What is the margin of error, assuming a 95% confidence level?

Example. Nine hundred (900) high school freshmen were randomly selected for a national survey. Among survey participants, the mean grade-point average (GPA) was 2.7, and the standard deviation was 0.4. What is the 95% confidence level?

Review

· Confidence intervals allow us to predict a population parameter based on a sample statistic.

· The population parameter we concentrated on was (population mean).

· The formula for the margin of error is

Module 03 - Part 2

Fill in the blank with what the variable represents in statistics.

1. - _________________________________

2. - _________________________________

3. - _________________________________

4. - _________________________________

5. - _________________________________

· _________________________________________ give us a range of possible values for the population parameter given a point estimate.

How to calculate a confidence interval:

· The _______________________________________________ is calculated by taking the critical value and multiplying it by the standard error of the statistic .

How to calculate the critical value depends on whether or not we are given the population standard deviation ().

· If the population standard deviation is known, use the ____________________.

· If the population standard deviation is unknown, use the ________________________.

When is unknown , we use the following method to calculate the margin of error:

· Use degrees of freedom: __________________________

· Calculate the t-statistic using ___________________________ in excel.

Example. Suppose we want to estimate the average weight of an adult male in Dekalb County, Georgia. We draw a random sample of 1,000 men from a population of 1,000,000 men and weigh them. We find that the average man in our sample weighs 180 pounds, and the standard deviation of the sample is 30 pounds. What is the 90% confidence interval?

Example. Suppose you work for the Department of Natural Resources and you want to estimate, with 95% confidence, the mean (average) length of all walleye fingerlings in a fish hatchery pond. You take a random sample of 10 fingerlings and determine that the average length is 7.5 inches and the sample standard deviation is 2.3 inches. What is the 95% confidence interval?

Example. Listed below are weights (hectograms or hg) of randomly selected girls at birth, based on data from the National Center for Health Statistics. Use the sample data to construct a 99% confidence interval for the mean birth weight of girls.

33 28 33 37 31 32 31 28 34 28 33 26 30 31 28

Let’s switch gears slightly. Now, we are going to look at examples where the population standard deviation, , is known. For this type of problem, we switch from a t-statistic to a z-score.

When is known, we use the following method to calculate the margin of error:

We could use NORM.INV to calculate our z-values, but since we don’t have to take the sample size into consideration (there are no degrees of freedom), we can simply pull the z-value from a chart:

Now we are going to switch from discussing confidence intervals to the size of the sample we are choosing! If we want to collect a sample to be used for estimating a population mean, how many sample values do we need?

To calculate the required sample size for a specific margin of error and population standard deviation, use the following formula.

Example. Assume that we want to estimate the mean IQ score for the population of statistics students. How many statistics students must be randomly selected for IQ tests if we want 95% confidence that the sample mean is within 3 IQ points of the population mean and we know the population standard deviation is 5 IQ points?

Example. A tax assessor wants to assess the mean property tax bill for all homeowners in Madison, Wisconsin. A survey ten years ago got a sample mean of $1400 and a standard deviation of $1000. How many tax records should be sampled for a 95% confidence interval to have a margin of error of $100?

Review

Confidence intervals have a different critical value depending on if the population standard deviation is known or not.

· If the population standard deviation is known, use the z-score.

· If the population standard deviation is unknown, use the t-statistic.

When trying to find the required sample size for a particular margin of error, we use the following formula

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