Chapter 8

Finding the Critical t-value for a Confidence Interval

- note confidence interval - 1 minus confidence level to find tails - find value on table

Finding the Value of t Given Area between −tα/2 and tα/2

-100% - percent will be the two tails - then view table and note info given in problem and answer found in first step. there will be a positive and negative

Using the Standard Normal Distribution to Estimate a Population Mean

-All possible samples of a given size have an equal probability of being chosen; that is, a simple random sample is used. -The population standard deviation, σ, is known. -Either the sample size is at least 30 (n≥30) or the population distribution is approximately normal.

Constructing a confidence interval with a given margin error

-Note the sample size, sample mean (also the point estimate), and margin of error -Find the lower end point. Lower endpoint = point estimate minus margin of error. -Find the upper end point. Upper end point = point estimate plus margin of error. -Write confidence interval in inequality notation or interval notation

Finding the Margin of Error of a Confidence Interval for a Population Mean (σ Known)

-Subtract 1 from confidence level -Divide that answer in half to get each tail -The critical value is the z score that has an area of 1 - confidence level to the right. -the negative of the critical value is the z score that has an area of 1 - confidence level to the left -use table to find z score of critical value -to get margin of error = multiply the z score by standard deviation/ square root of sample size

Constructing a Confidence Interval for a Population Mean (σ Unknown)

1) Find the point estimate 2) Find the margin of error 3) Subtract the margin of error from and add the margin of error to the point estimate

Constructing a Confidence Interval for a Population Mean (σ Known)

1) Find the point estimate. 2) Finding the Margin of Error of a Confidence Interval for a Population Mean (σ Known) 3) Subtract the margin of error from and add the margin of error to the point estimate.

Finding the Margin of Error of a Confidence Interval for a Population Mean (σ Unknown)

1) Note confidence level. Alpha is 1- confidence level. each tell is alpha divided by 2 2) Find critical value by looking at table 3) Note standard deviation 4) Note sample size 5) critical value * standard deviation / square root of sample

Minimum Sample Size for Estimating a Population Mean

1)Find confidence level then subtract one from the confidence level. (Ex 90% of .90, 1-.90=.1) then divide that number in half. (.1/2=.005) 2)Then find z score by looking up answer on table of critical values 3)find margin of error 4) find standard deviation 5) answer equals critical value * standard deviation / margin of error then the answer to that to the power of 2

confidence interval

1- confidence level/2

How will decreasing the level of confidence without changing the sample size affect the width of a confidence interval for a population mean? Assume that the population standard deviation is unknown and the population distribution is approximately normal.

The margin of error will decrease because the critical value will decrease. The decreased margin of error will cause the confidence interval to be narrower.

How will increasing the level of confidence without changing the sample size affect the width of a confidence interval for a population mean? Assume that the population standard deviation is unknown and the population distribution is approximately normal.

The margin of error will increase because the critical value will increase. The increased margin of error will cause the confidence interval to be wider.

Student's t-Distribution

a probability distribution for a continuous random variable, X, defined completely by its number of degrees of freedom, such that the following properties are true: 1) A t-distribution curve is symmetric and bell-shaped, centered about 0. 2)A t-distribution curve is completely defined by its number of degrees of freedom, df. 3)The total area under a t-distribution curve equals 11. 4)The x-axis is a horizontal asymptote for a t-distribution curve.

interval estimate

a range of possible values for a population parameter.

point estimate

a single-number estimate of a population parameter.

df=

degrees of freedom

level of confidence

denoted by c, is the percentage of all possible samples of a given size that will produce interval estimates that contain the actual parameter.

Critical z-values

denoted −zα/2 and zα/2, mark the boundaries for the area under the middle of the standard normal curve that corresponds with a particular level of confidence, c.

margin of error

or maximum error of estimate, E, is the largest possible distance from the point estimate that a confidence interval will cover.

unbiased estimator

point estimate that does not consistently underestimate or overestimate the population parameter.

Finding a point estimate for a population mean

x bar = add all data values/# of data values or if given it is the sample mean