Confidence Intervals

Warm-up You are wanting to find the average number of siblings a student at Iowa State has. Instead of taking a census, you decide to obtain a sample and use that sample to estimate the true average. Voluntary Response Convenience Sample Systematic Sample

Conditions You have a simple random sample Sample size is large np < 10 n(1-p) < 10 ^ ^

Confidence interval Math symbol Def’n How to find it 𝑡 𝛼 2 𝑡 𝛼 2 Look up in table Typically use 1.96 for 95% CI confidence level Given to you 𝑋 Estimated mean se( 𝑋 ) Estimated standard error Degrees of freedom

Confidence interval Math symbol Def’n How to find it 𝑡 𝛼 2 𝑡 𝛼 2 T-distribution related to 𝛼 Look up in table Typically use 1.96 for 95% CI 𝛼 confidence level Given to you 𝑋 Estimated mean 𝑥 𝑛 se( 𝑋 ) Estimated standard error 𝑠 𝑛 df Degrees of freedom n-1 ^

𝑋 - 𝑡 𝛼 2 (𝑠𝑒( 𝑋 )) to 𝑋 + 𝑡 𝛼 2 (𝑠𝑒( 𝑋 ))

A package of light bulbs promises an average life of more than 750 hours per bulb. A consumer group did not believe the claim and tested a sample of 40 bulbs. The average lifetime of these 40 bulbs was 740 hours with s=30 hours. Find the consumers groups 95% confidence interval Based on this, what conclusions would you make about the promise of 750 hours. If the manufacturer follows the same estimate of s, what is their 95% confidence interval Based on this, what might their response be if the consumer group complains they lie.

Create a generic sentence that can apply to any confidence interval in any situation. (Leave blanks where you would specify when given context)

Interpret I am 95% confident that the mean _______ lies between _____ and _____. 95% of the confidence intervals made this way would contain the true population mean

A catalog sales company promises to deliver orders placed on the Internet within 3 days. Follow-up calls to randomly selected customers show that a 95% confidence interval for the proportion of all orders that arrive on time is 88% +/- 6%. What does this mean? Which of these are correct Between 82% and 94% of all orders arrive on time 95% of all random samples of customers will show that 88% of orders arrived on time 95% of all random samples of customers will show that 82% to 94% of orders arrived on time. We are 95% sure that between 82% and 94% of the orders placed by the customers in this sample arrived on time On a randomly chosen day, we can be 95% confident that between 82% and 94% of the large volume of orders will arrive on time

A catalog sales company promises to deliver orders placed on the Internet within 3 days. Follow-up calls to randomly selected customers show that a 95% confidence interval for the proportion of all orders that arrive on time is 88% +/- 6%. What does this mean? Which of these are correct Between 82% and 94% of all orders arrive on time 95% of all random samples of customers will show that 88% of orders arrived on time 95% of all random samples of customers will show that 82% to 94% of orders arrived on time. We are 95% sure that between 82% and 94% of the orders placed by the customers in this sample arrived on time On a randomly chosen day, we can be 95% confident that between 82% and 94% of the large volume of orders will arrive on time