Collection and Use of Data in Economics

 Point Estimate: single number

 Confidence Interval Estimate: range of numbers

 95% or (1 - ) where  is the area outside the tails.

 The general form for all confidence intervals is:

Point Estimate ± Margin of Error

Point & Confidence Interval Estimates

Population Mean Estimation (σ Known)  Assumptions

 Population σ is known

 Population is normally distributed (or n is >30)

 Confidence Interval Estimate:

(where z/2 is the normal distribution value for a probability of /2 in each tail)

 End points:

 Margin of error can be reduced if:  Population standard deviation is reduced (σ↓)

 Sample size is increased (n↑)

 Confidence level is decreased, (1 – ) ↓

n

σ zx α/2



n

σ zxUCL α/2

 n

σ zxLCL α/2



MEx →

Application

1. A random sample of 75 students taking the quantitative analysis test at SDSU

showed an average score of 18.3 with a population standard deviation of 5.0.

What would be a 95% confidence interval for all scores on the quant?

2. A sample of 120 randomly selected cars showed highway speeds in a certain

location that averaged 66.2 MPH with a population standard deviation of 13.8

MPH. Find a 99% confidence interval for the average highway speed of the cars

at this location.

3. A recent study of 38 college students found they spent an average $275 per

semester for books with a population standard deviation of $28. Find a 90%

confidence interval for the average amount spent per semester by college

students.

 Student’s t distribution

 Family of distributions that varies with the degrees of freedom:

 Confidence Interval Estimate:

 End points:

 Use t instead of Z when σ is unknown and n<30

d.f. = n - 1

Population Mean Estimation (σ Unknown)  Substitute σ with the sample standard deviation s.

 This introduces extra uncertainty, since s varies from sample to sample so we

use the t-distribution instead of the normal distribution.

n

s tx

α/21,-n  MEx →

n

σ txUCL

α/21,-n 

n

σ txLCL

α/21,-n 

Student’s t-Table

Application 1. A medical researcher would like to estimate the average resting heart rate of

subjects that are being treated with a new medication. A sample of size n=18

gives an average resting rate of 87.2 with a sample standard deviation of 11.31.

Find a 90% confidence interval for the true resting rate.

2. The average deposit of 20 customers selected at random from the depositors of

a local bank is $83.60 with a sample standard deviation of $12.41. What would

be a 95% confidence interval for the mean deposit of all bank depositors?

3. A researcher would like to estimate the average weight loss of people using a

new high protein diet. A random sample of 12 people using the diet showed an

average weight loss of 12.3 pounds with s = 2.5 pounds. Find a 90%

confidence interval for the true mean weight loss.

Application  For each of the following identify the appropriate distribution to use, t or Z, in inferring sample

statistics to population parameters.

1. The average time it took a sample of 45 selected mice to complete a certain

maze was 3.2 minutes. The population standard deviation is unknown but the

sample standard deviation is 0.4 minutes.

2. A sample of 20 sport fishing boats working out of San Diego ports showed an

average catch of 80 tuna on three day trips during tuna season. The population

standard deviation is known to be 12.8 fish.

3. A random sample of 18 families found that they sent an average of 44.7 greeting

cards during the holiday season with a sample standard deviation of 8.1.

Population Proportion Estimation  The distribution of the sample proportion is approximately normal if the sample size is large.

 Normal can be used to approximate the binomial when: nP and n(1−P) > 5

 Population standard deviation can be estimated:

 Confidence Interval Estimate:

 Endpoints of interval are:

n

)p̂(1p̂  n

P)P(1 σ

P

 

n

)p̂(1p̂ zp̂ α/2

 

n

)p̂(1p̂ z-p̂LCL α/2

 

n

)p̂(1p̂ zp̂UCL α/2

 

→

→ Ep̂ M

Application 1. A Cardiologist is doing a study of damage to the heart muscle due to heart

attack. As part of the study, the doctor would like to know the proportion of

patients who suffer a second heart attack within one year of their first attack. A

random sample of 300 patients finds that 64 suffer a second heart attack within

one year. Find a 98% confidence interval for the true proportion of patients who

suffer a second heart attack within one year of their first.

2. To estimate the number of accidents that involve alcohol, 80 past accidents

were chosen at random. Sixty three were found to involve alcohol. Find a 90%

confidence interval for the true proportion of accidents that involve alcohol.

3. A survey asked college students the following question. Do you consider

yourself a social liberal? The survey questioned 220 students and 168

answered yes to the question. Find a 94% confidence interval for the true

proportion of students who consider themselves social liberals.