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