AP Statistics

AP Statistics10.1 Estimating With Confidence

Calculate and interpret a confidence interval

Know the difference between confidence level and confidence interval

Understand how standard deviation, sample size, and the critical value affects the

margin of error

Find the sample size needed for a given margin of error

Learning Objectives:

What are the basic facts about the sampling distribution of ?

1- has a normal distribution.  2- the mean of this normal dist. Is the same

as the true mean (μ).

3- the standard deviation of is σ/√n.

Review:

Suppose I randomly chose a sample of 25 students test scores. The mean test score was 87 and the standard deviation for the population of students who have taken this test is 10.

What is the standard deviation of ? 10/√25=2

What does this sampling distribution look like?

N(87,2)

In formal inference we use confidence intervals to express the strength of our conclusions about a population from our sample data.

When we use statistical inference you are acting as if data were a random sample or comes from a randomized experiment.

Confidence interval

Confidence interval-

Margin of error-

Formulas:

Ex 1: 1025 women were interviewed. 47% of the women said they did not get enough time for themselves.

a)The poll announced a margin of error of +/- 3 percentage points for 95% confidence in its conclusion. What is the 95% confidence interval for the percent of all adult women who think they don’t get enough time for themselves?

47% +/- 3%=(44%,50%)

b)Explain to someone who knows nothing about statistics, why we can’t just say that 47% of all adult women do not get enough time for themselves. Then explain clearly what “95% confidence” means?

We are 95% confident that the true mean of adult women who do not get enough time for themselves falls between 44% and 50%.

Ex 2: A student reads that a 95% confidence interval for the mean NAEP quantitative score for men of ages 21 to 25 is 267.8 to 276.2. When explaining, a student said “95% of all young men have scores between 267.8 and 276.2.” Is this correct and why?

No-we are 95% confident the true mean falls between 267.8 and 276.2.

A level C confidence interval for a parameter is an interval computed from sample data by a method that has probability C of producing an interval containing the true value of the parameter.

Confidence Intervals:

Confidence Level

Tail Area Z*

90% 0.05 1.64595% 0.025 1.96099% 0.005 2.576

Results for common confidence levels.

The number z* with probability p lying to its right under the standard normal curve is called the upper p critical value of the standard normal distribution.

How to find the level C confidence interval:99% c=0.99 95% c=0.95 90% c=0.90

Critical Values:

Ex: A pharmaceutical company analyzes a specimen from each batch of a product to verify the concentration of the active ingredient. The chemical analysis is not perfectly precise. Repeated measurements on the same specimen give slightly different results. The results of the repeated measurements follow a normal distribution very closely. The procedure has no bias, so the mean ( μ ) of the population of all measurements is the true concentration in the specimen. The standard deviation of this distribution is known to be σ =.0068 grams per liter. The lab analyzes each specimen 3 times and reports the mean result.

3 analyses of one specimen give concentrations .8403 .8363 .8447

We want a 99% confidence interval for the true concentration u.

One sample z intervalAssumptions:-random sample-normal distribution(with no outliers)0.8404 +/- 2.576(0.0068/√3)=

(0.8303,0.8505)

Interpret this interval. We are 99% confident that the true mean

specimen measurement is between 0.83 and 0.85.

Find the difference between a 99% and a 90% confidence interval:

The 99% confidence interval is wider!!

When estimating a parameter, we want high confidence with a small margin of error.

Margin of error :

What happens to the margin of error when you change:

#1) n (your sample size)increase sample, margin of error is smaller

#2) σ decrease σ, margin of error is smaller

#3) C (your confidence level) decrease C, margin of error is smaller

Characteristics of Confidence Intervals

To determine the sample size n for a specified level of confidence using the margin of error m,

  *** Remember, n is always a whole number,

so you need to round up****

Choosing Sample Size:

Example: A school wants to estimate the number of hours students spent studying in a week to within 15 minutes with a 95% confidence level. How many students would be questioned if the standard deviation was 45 minutes?

n=34.57 therefore n=35.

When we say something is normally distributed, we have to take into account the sample size.

n≥30 (n is large)

n is between 15-29(with no extreme outliers)

n is less than 15 (with no outliers)

Assumptions: