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AP Statistics Chapter 10: Estimating with Confidence

10.1 – Confidence Intervals: The Basics

Statistical Inference
Statistical inference provides methods for drawing conclusions about a population from sample data.

Confidence Interval

A level C confidence interval is of the form

estimate ± margin of error

The confidence level C gives us the probability that the interval will capture the true mean of the population. So when we use a 95% confidence level, we have a 95% chance of arriving at an interval containing the true population mean.

Conditions for Constructing a Confidence Interval for a Population Mean
The construction of a confidence interval for a population mean is appropriate when

  • The data come from an SRS from the population of interest
  • The sampling distribution of x-bar is approximately normal*
* This can be achieved if n is large (at least 30) or if our statistic already adheres to a normal distribution (then, any sample size is OK)

Confidence Interval for a Population Mean

The form of the confidence interval for a population mean when the population standard deviation is known is  

z_interval

Sample size for a desired margin of error

To determine the sample size (n) for a given margin of error m in a sample mean t interval use the formula

sample_size_mean

Remember, that we will always round up to ensure a smaller margin of error.


10.2 – Estimating a Population Mean

Conditions for Inference about a Population Mean

  • SRS - Our data are a simple random sample (SRS) of size n from the population of interest. This condition is very important.
  • Normality - Observations from the population have a normal distribution with mean mu and standard deviation sigma. In practice, it is enough that the distribution be symmetric and single-peaked unless the sample is very small. Both mu and sigmaare unknown parameters.
  • Independence - The population size is at least 10 times greater than the sample size

Standard Error

When the standard deviation of a statistic is estimated from the data, the result is called the standard error of the statistic. The standard error of the sample mean is

std_error

The form of the confidence interval for a population mean with n-1 degrees of freedom is

t_interval

To compare the responses to the two treatments in a paired data design, apply the one-sample t procedures to the observed differences.

For example, suppose that pre and post test scores for 10 individuals in a summer reading program are:

Subject

1

2

2

4

5

6

7

8

9

10

Pre-test

25

31

28

27

30

31

22

18

24

30

Post-test

28

30

34

35

32

31

26

16

28

36

Difference

3

-1

6

8

2

0

4

-2

4

6

We would use the data in the differences row and perform one-sample t analysis on it. t Procedes

  • Except in the case of small samples, the assumption that the data are an SRS from the population of interest is more important than the assumption that the population distribution is normal.
  • Sample size less than 15. Use t procedures if the data are close to normal. If the data are clearly nonnormal or if outliers are present, do not use t.
  • Sample size at least 15. The t procedures can be used except in the presence of outliers or strong skewness.
  • Large samples. The t procedures can be used even for clearly skewed distributions when the sample is large, roughly n≥30.

 

10.3 – Estimating a Population Proportion

Conditions for Inference about a Population Proportion
  • SRS - The data are a simple random sample (SRS) from the population of interest.
  • Normality - Counts of successes and failures must be 10 or more.
  • Independence - The population size is at least 10 times greater than the sample size

Standard Error

When the standard deviation of a statistic is estimated from the data, the result is called the standard error of the statistic. The standard error of the sample proportion p-hat is

  p-hat_SE

The form of the confidence interval for a population proportion is

OnePropZInt    

Sample size for a desired margin of error

To determine the sample size (n) for a given margin of error m in a 1-proportion z interval, use the formula

samp_size_Prop

where p* = 0.5, unless another value is given.

Remember, that we will always round up to ensure a smaller margin of error.

 

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