Chapter 7

The critical value z alpha/2 is the positive zvalue that is at the vertical boundary separating an area of

alpha/2 in the right tail of the standard normal distribution. (The value of -z alpha/2 is at the vertical boundary for the area of alpha/2 in the left tail.)

A z score associated with a sample proportion has a probability of

alpha/2 of falling in the right tail.

The z score separating the right -tail region is commonly denoted by z alpha/2 , and is referred to as a critical value because it is on the

borderline separating z scores from sample proportions that are likely to occur from those that are unlikely to occur.

The margin of error E is also called the maximum error of the estimate and can be found by multiplying the

critical value and the standard deviation of sample proportions. E= Z alpha/2 Sqrt (p hat times q hat/ n)

It is unbiased in the sense that the distribution of sample proportions tends to center about the value of

p, that is, sample proportion p with a hat do not systematically tend to underestimate or over estimate p.

The sample mean is the best point estimate of the

population mean μ.

Confidence Level : The confidence level is the probability 1-α ( often expressed as the equivalent percentage value) that the confidence interval actually does contain the

population parameter, assuming that the estimation process is repeated a large number of times. ( The confidence level is also called the degrees of confidence, or the confidence coefficient. )

A sample proportion p with a hat is the best estimate of the

population proportion p.

Point Estimate : The sample proportion p hat is the best point estimate of the

population proportion p.

A confidence level of 95% tells us that the process we are using will, in the long run result in confidence interval limits that contain the true population

proportion 95% of the time.

Because a point estimate has a serious flaw of not revealing anything about how good it is, statisticians have cleverly developed another type of estimate . This estimate , called a confidence interval or interval estimate, consists of a

range ( or an interval) of values instead of just a single value

2. When the original set of data is unknown and only the summary statistics ( n, xs) are used, round the confidence interval limits to the

same number of decimal places use for the sample mean.

The normal distribution is used as the distribution of

sample means.

Critical Value: A critical value is the number on the borderline separating

sample statistics that are likely to occur from those that are unlikely to occur.

It is the most consistent estimator in the sense that the standard deviation of sample proportions tends to be smaller than the

standard deviation of any other unbiased estimators.

Confidence Level:: The confidence interval associated with a confidence level, such as 0.95 (or 95%). The confidence level gives us the

success rate of the procedure used to construct the confidence interval. α is the complement of the confidence level. For a 0.95 ( or 95%) confidence level, α=0.05 and z alpha/2=1.96.

After obtaining the confidence interval estimate of the population mean μ such as a 95% confidence interval of 164.49 < μ < 180.61, there is a correct interpretation and many incorrect interpretations. Correct:

" We are 95% confident that the interval from 164.49 to 180.61 actually does contain the true value of μ".

The most common choices of the confidence level are

90% (with α=0.10), 95% (with α=0.05) and 99% ( with α= 0.01).

We are 95% confident that the interval from 0.677 to 0.723 actually does contain the true value of the population proportion p." This means that if we were to select many different samples of size 1501 and construct the corresponding confidence intervals,

95% of them would actually contain the value of the population proportion p.

For a 95% confidence level, α=0.05, so there is a probability of

0.05 that the sample proportion will be in error by more than E.

The choice of 95% is most common because it provides a good balance between precision and reliability. The 0.95 ( or 95%) confidence interval estimate of the population proportion p is

0.677<p<0.723.

Two major activities of inferential statistics :

1) To use sample data to Estimate values of population parameters (such as a population proportion or population mean ) and 2) to test hypothesis or claims made about population parameters.

Point Estimate:

A point estimate is a single value (or point ) used to approximate a population parameter.

The number z alpha/2 is a critical value that is a z score with the property that it separates an area of

alpha/2 in the right tail of the standard normal distribution.

The subscript alpha/2 is simply a reminder that the zscore separates an area of

alpha/2 in the right tail of the standard normal distribution.

Descriptive Statistics :

Is used when we summarize data using tools such as graphs, and statistics such as mean and standard deviation.

Inferential Statistics:

Is used when we use sample data to make inferences about population parameters.

Confidence Interval : A confidence interval ( or interval estimate) is a range ( or an interval) of values used to

estimate the true value of a population parameter. A confidence interval is sometimes abbreviated as CI.

Although sample mean is the best point estimate of the population mean , it does not give us any indication of

how good our best estimate is. We get more information from a confidence interval ( or interval estimate), which consist of a range ( or an interval ) of values instead of just a single value.

Critical Values: A standard z score can be used to distinguish between sample statistics that are

likely to occur and those that are unlikely to occur. Such a z score is called a critical value.

When data from a simple random sample are used to estimate a population proportion p, the margin of error, denoted by E, is the

maximum likely difference ( with probability 1-α, such as 0.95) between the observed sample proportion and the true value of the population proportion p .

If the computed sample size n not a whole number, round the value of n up to the

next larger whole number.

Under certain conditions, the sampling distribution of sample proportions can be approximated by a

normal distribution.

If the original population is not itself normally distributed, we can say that means of samples with size n>30 have a distribution that can be approximated by a

normal distribution. Sample sizes of 15 to 30 are sufficient if the population has a distribution that is not far from normal, but some other populations have distributions that are extremely far from normal and sample sizes greater than 30 might be necessary . Here we use the simplified criterion of n>30 as justification for treating the distribution of sample mean as a normal distribution.

When using the original set of data to construct a confidence interval , round the confidence interval limits to

one more decimal place than is used for the original set of data.

The sample mean is an unbiased estimator of the population mean μ, and for many populations, sample means tend to vary less than

other measures of center, so the sample mean is usually the best point estimate of the population mean μ.

p with a hat is used as the point estimate of p because it is

unbiased and it is the most consistent estimator.