Quiz 2 - Biostats - Unit 7: Hypothesis Test

Ex: A cigarette manufacturer claims that the average nicotine content of a brand of cigarettes is at most 1.5 milligrams

left tailed test (<)

A hypothesis test

(aka test of significance) is a standard decision-making procedure for testing the statement about a POPULATION

Critical Value (s)

-The critical value(s) separates rejection region from the non-rejection region -Critical value(s) indicates that the null hypothesis should be rejected when the test statistic is in the rejection region on one side of the mean -Critical values (kind of cut-off value) for an significance level test denoted as: - one sided test: Z significance level - two sided test: Z significance level

2. Test statistic for population mean (small samples) -> t

-When the population standard deviation is unknown and n < 30, the z - test is inappropriate for testing hypotheses involving mean -The t-test is used in this case t = x bar - mu/s /squared root n Where: X bar = sample mean mu : hypothesized population mean s = sample standard deviation n : sample size degrees of freedom = n-1

What is statistically significant?

-When you run an experiment or analyze data, you want to know if your findings are significant -Statistical significance DOES NOT mean important, it means THERE IS EVIDENCE THAT THE EFFECT WE OBSERVED IN SAMPLE EXISTS IN POPULATION -When a finding is significant, it simply means you can FEEL CONFIDENT THAT'S IT REAL -> VERY UNLIKELY THAT HAPPENED BY CHANCE *Something can be statistically significant but small and unimportant

P-value approach

-besides listing an signignifance value, many computer statistical packages give a P-value for hypothesis tests -After calculating a test statistic, the software converts the test statistic to a p-value under the null hypothesis -We use p-value to decide a test is statistically significant

Significance level

-common choices for signicance level are 0.05, 0.01, and 0.10 -(1-significance level) * 100% = confidence level, for a given significance level value -The significance level (denoted by significance level) determines THE SIZE OF REJECTION REGION

Steps for hypotehsis test

1) set up null and alternative hypothesis 2) Determine critical value(s) 3) Collect sample and summarize the data into a statistic 4) decide whether the result is statistically significant

Step 3: Collect sample and summarize the data into a statistic Step 4: Decide whether the result is statistically significant

1. Collect sample information 2. Compute sample mean (point estimate) 3. Transform sample mean to a test statistic (z or t): STANDARDIZED VERSION OF THE POINT ESTIMATE 4. Decide whether the null hypothesis should be rejected Compare test statistic with critical value 1. If z or t statistic falls in the rejection region -> Reject the null hypothesis 2. If z or t statistic falls outside the rejection region -> Fail to reject the null hypothesis

What is a hypothesis test

Hypothesis is a statment about an unknown POPULATION parameter (e.g. proportion, mean, or standard deviation of the population)

Ex: Among 157 AA mean, the mean systolic blood pressure was 146 mmHg with a standard deviation of 27 mmHg. We wish to know if based on these data, we ay conclude that the mean systolic blood pressure for a population of AA is greater than 140. Use significance level = 0.01. H0 = mu = 140 H1 = mu > 140 (claim)

P value is 0.0027 and since the p value is less than 0.01 the decision is to reject the null hypothesis Interpretation: There is enough evidence to support the claim that the mean systolic blood pressure for a population of AA is greater than 140 mmHg

Ex: Many people think that "normal" body temp is lower than 98.6. Is that true at significance level = 0.05. Data: 16 donors at a blood bank, under age 30. sample mean = 98.2 degrees, s = 0.497 degrees Ho = mu = 98.6 H1 = mu < 98.6 (claim)

P value is 0.00286 so since the p value is less than 0.05, the decision to reject the null hypothesis Interpretation: there is enough evidence to support the claim that the population mean body temperature is less than 98.6

Ex: A researcher thinks that if expectant mothers use vitamins, the babies' birth weight will increase. The average birth weight of the population is 8.6 pounds.

Right tailed (>)

P-values -decision rule

The p value is the area underneath the propbabaility curve at or more extreme of a calculated test statistic A) If p value < or equal to significance level -> reject the null hypothesis B) If p value > to significance level -> do not reject the null hypothesis

1. Test statistic for population mean (Large Samples) -> Z

The z test is a statistical test of the mean of a population. It can be used when n > or equal to 30 or when the population is normally distributed and standard deviation is known z = x bar - mu / standard deviation / squared root n Where: x bar: sample mean mu : hypothesized population mean standard deviation : population standard deviation n : sample size magnitude of difference between sample mean and population mean

Right tailed (>)

When the 'alternative hypothesis' states that the parameter is greater than, larger than, or above the null value

Left tailed (<)

When the alternative hypothesis states that the parameter is smaller, less than, or below the null value

Purpose:

Whether the sample mean (x bar) is significantly greater, less, or different than any hypothetical population mean (mu) -Based on the collected sample, we would like to make a decision about the population research hypotehsis: Reject H0 or fail to reject H0 You should NOT say: "H0 is accepted"

A rejection region

is an area of a graph where you will reject the null hypthesis if your test falls inot that area

Ex: A researcher reports that the average salary of college professors in the US is more than $62,000. A sample of 30 college professors has a mean salary of $63,260. At significance level = 0.05, test the claim that college professors earn more than $62,000 a year. The standard deviation of the population is $5230

since the test value, + 1.32 is less than the ciritical value, +1.65, and not in the ciritical region, the decision is Fail to reject the null hypothesis Interpretation: There is not enough evidence to support the claim that college professors earn more on average than $62,000 a year at a 95% confidence level

P-value

tells you HOW LIKELY it is that your data could have occurred UNDER THE NULL HYPOTHESIS (chance alone)

Ex: We would like to test whether the pulse rate ina group will differ from the mean popualtion pulse rate of 82 beats per minute

two-tailed test

Two tailed:

when the alternative hypothesis states that the parameter is not equal to or is different from the null value