exam 3

T score vs Z score

* NEVER use t distribution for estimating CI for the proportion

Steps in Hypothesis Testing

1. State the null and alternative hypotheses 2. Specify the level of significance a 3. Collect the sample data and compute the test statistic Critical value approach: 4. Use the level of significance 𝛼 to determine the critical value 5. Compare the value of the test statistic and the critical value to determine whether to reject H0 p-value approach: 4. Use the value of the test stat to find the p-value 5. compare p-value and a: reject if p-value<a, do not reject if p-value ≥ 𝛼

increasing the confidence level

1. increases the critical value/critical z-score (z a/2) 2. wider confidence interval 3. Margin of error increases 4. area in the middle of the z distribution increases

Point ESTIMATE

A particular value of the estimator obtained from the sample (the value of a sample statistic derived from a specific sample) - xbar = $54,000 is the estimate of the population mean mu - s = $100 is the estimate of the population standard deviation sigma - 𝑝bar = 0.8 is the estimate of the population proportion p

Point ESTIMATOR

A statistic used to estimate a population parameter (a more general term - it's a statistic uses to obtain the point estimate for a given sample data) - xbar is the point estimator of the population mean mu - s is the point estimator of the population standard deviation sigma - 𝑝bar is the point estimator of the population proportion p

null hypothesis

H0, is a presumed default state of nature / status quo / prior belief to be challenged in the testing procedure *≤, =, or ≥

Hypothesis Testing for the Population Mean, SIGMA IS KNOWN

In order to implement the test, it is essential that the sampling distribution of XBAR is normal: 1. If the sample size is small (n < 30) the population must follow the normal distribution 2. If the sample size is large (n ≥ 30) the Central Limit Theorem states that the sampling distribution of the mean follows the normal distribution (so, there is no restriction on the population distribution)

ASSUMPTIONS (CI FOR PROPORTION)

Sampling distribution of pbar is normal np ≥ 5 and n(1 - p) ≥ 5 Confidence interval for the population proportion may or may not contain the population proportion

ASSUMPTIONS (CI FOR MEAN, SIGMA KNOWN)

Sampling distribution of ҧ 𝑥 is normal The sample size is at least 30 (n ≥ 30) The population distribution is norma

ASSUMPTIONS (CI FOR MEAN, SIGMA UNKNOWN)

The sample size is at least 30 (n ≥ 30) The population distribution is normal

degrees of freedom

The shape of the curve depends on the degrees of freedom (df), df = n - 1 • The degrees of freedom determine the extent of the broadness of the tails: the fewer the degrees of freedom, the broader the tails

What is the purpose of calculating a confidence interval for the proportion?

To provide a range of values that, with a certain measure of confidence, contains the population proportion.

Confidence Interval for the mean, SIGMA IS UNKNOWN

When the population standard deviation 𝜎 is unknown, we substitute s, the sample standard deviation, in its place to estimate the standard error of the mean

Confidence Interval for the mean, SIGMA IS KNOWN, SMALL SAMPLE

When the sample size n is less than 30 and 𝜎 is known, the population must be normally distributed to calculate a reliable confidence interval - Confidence intervals for the mean require that the 𝑥 is normal - sampling distribution is always normal (regardless of the sample size) if the population data is normally distributed

null hypothesis can NEVER

be accepted. We can only reject, or fail to reject based on: 1) The sample result provides enough evidence to reject H0 2) The sample does not provide enough evidence to reject H0

z a/2

critical z score 𝑧 𝛼/2 is the z value that corresponds to the area of 𝛼/2 in the right tail of the standard normal distribution

alternative hypotheses

denoted by H1, is the statement opposite (complementary) to the null hypothesis *>, ≠, or <

Hypothesis Testing for the Population Proportion

ex: Whether more than 50% of physicians have been sued due to malpractice? • Whether the proportion of those who receive coupons in the stores and actually use them later is greater than 10%? • Whether the proportion of employers who gave holiday gifts to their employees is greater than 35%? • Whether the proportion of those who live in their state of birth has increased?

Confidence interval for the mean

interval estimate around a sample mean that provides us with a range within which the true population mean is expected to lie

Confidence Intervals for Proportions

is an interval estimate around a sample proportion that provides us with a range of values within which the true population proportion is expected to lie

Type II error

occurs when we fail to reject the null hypothesis when it is not true denoted by β

Interval estimate

provides a range of values that best describes the population parameter of interest

Increasing the sample size while keeping the confidence level constant...

reduces the margin of error, resulting in a narrower (more precise) confidence interval

As the number of degrees of freedom increases

shape of the t-distribution becomes similar to the normal distribution *With more than 100 degrees of freedom (a sample size n of more than 100), the two distributions are practically identical

a

significance level a = 1 - Confidence Level

Student's t-distribution

the Student's t-distribution is used in place of the standard normal distribution to find the critical value Only used in terms of the mean

Margin of Error

the amount we add and subtract to the point estimate to form the confidence interval

p-value

the probability of observing a sample mean at least as extreme as the one selected for the hypothesis test, assuming the null hypothesis is true with equality

one tail Testing for the Population Mean, SIGMA IS UNKNOWN

use t score

one-tail hypothesis test

used when the alternative hypothesis is stated as < or > Upper Tail: H1> Lower Tail: H1<

two-tail hypothesis test

used whenever the alternative hypothesis is expressed as ≠

Hypothesis Testing for the Population Mean, SIGMA IS UNKNOWN

we substitute the sample standard deviation, s, in place of 𝜎 We use the Student's t-distribution with df = n - 1 to find the critical value and the p-value rather than the standard normal distribution Hypothesis testing procedure is valid only if the sampling distribution of 𝑥bar is normal: • The population data follow the normal distribution • Sample size is large (n ≥ 30)

Type I error

when the null hypothesis is rejected when it is true denoted by a- level of significance

one tail Testing for the Population Mean, SIGMA IS KNOWN

z-test statistic for a hypothesis test for the population mean when 𝜎 is known