Stat 50 - Chapter 7

Degrees of freedom

n-1 The number of sample values that can vary after certain restrictions have been imposed on all data values

Critical value

the number on the borderline separating sample statistics that are likely to occur from those that atre unlikely. The number z(α/2) is a cricital value that is a z score with a property that it separates an area of α/2 in the right tail of the standard normal distribution.

Confidence level

the probability 1 - α that the confidence interval actually does contain the population parameter, assuming that the estimation process is repeated a large number of times (aka degree of confidence, confidence coefficient) Gives us the success rate of the procedure

q^ = ?

1 - p^

Properties of X² distribution

1. Not symmetric 2. Values can be 0 or positive (not negative) 3. Different for each # of df. As df increases, X² approaches a normal distribution

Conditions for using CI to estimate p

1. Simple random sample 2. Binomial distribution: fixed # of trials, trials indep, 2 categories of outcomes, prob's constant 3. There are at least 5 successes and 5 failures

Requirements for using CI to estimate population SD or variance?

1. simple random sample 2. population is normally distributed

Properties of student t distribution

1. t distribution is different for different sample sizes 2. More variability with small samples than standard normal distributions 3. mean is at t = 0 4. SD > 1 5. As n gets larger, the distribution gets closer to standard normal distribution

How to round confidence interval for proportion?

3 significant digits

Confidence interval

Abbreviated CI, a range of values used to estimate the true value of a population parameter.

How to round sample size?

Always round UP to next whole number

What does each area (top row) of X² table represents?

Cumulative area to the RIGHT of the critical value

Margin of error when σ is unknown

E = t(α/2) × s/√n

How to round CI for SD and variance?

One more decimal place than raw data

How to round confidence interval for the mean?

One more decimal place than raw data

How do we find the sample size when no estimate of p^ is known?

Replace p^ with 0.5 and q^ with 0.5. Therefore, p^q^ = 0.25

How to determine sample size to be used for estimating µ when σ is unknown?

Use range rule of thumb to estimate SD or calculate and use sample SD, s

Estimating a population mean when σ is not known

Use t distribution and t equation Requires normality or n > 30

Estimating a population mean when σ is known

Use the standard normal distribution (z scores) to construct confidence interval Requires normality or n>30

How to interpret the CI of 95% with 0.828 < p < 0.872.

We are 95% confident that the interval from 0.828 to 0.872 actually does contain the true value of the population proportion p.

Margin of error

When data from a simple random sample are used to estimate a population proportion p, E, is the maximum likely difference between the observed sample proportion p^ and the true value of the population proportion p. E is also called the maximum error of the estimate

Chi-Square distribution

X² = (n-1)s²/σ² X² is a sample statistic

Point estimate

a single value used to approximate a population parameter The sample proportion p^ is the best point estimate of the population proportion p (p^ is unbiased). p^ = X/n

Interval estimate

an interval or range of values used to estimate the parameter. this may or may not contain the parameter being estimated

What do we use to construct a confidence interval estimate of a population SD or Variance?

chi-square (X²)