FUNDAMENTALS OF INFERENTIAL STATISTICS

inferential statistics – drawing conclusions on the population based on the samples.parametric tests

examples:o z-testso t-testso Analysis of Variance (ANOVA)

non-parametric testsexamples:o Mann-Whitney U Testo Wilcoxon Signed Rank Testo Kruskal Wallis Testo Friedman Test

Recall that: population – the target datasample – a subset of population

group size: A group is considered large if n ≥ 30. Otherwise, small.Z – tests are usually used for large groups and t-tests for small groups.*z-tests and t-tests, oftentimes, can perform each other’s role in inferential statistics.

It only differs in setting the critical value.Remember the Central Limit Theorem.

Hypothesis – assumptionNull hypothesis (H0)- expresses that the parameter is equal to a hypothesized value or the parameters are all equal.- carries the equal sign: = Alternative hypothesis (Ha)- expresses that the parameter is greater or less than a hypothesized value or the parameters are not

equal (may be less or greater than the hypothesized value)- carries the inequality sign: < or > for one-tailed tests and ≠ for two-tailed tests

Two types of tests according to directionOne-tailed test (also called “directional test” or “one-directional test”)

- determines if the sample mean is from a population with mean that is significantly less or greater than the hypothesized mean.

- determines if a group is better or worse than the other.

Two-tailed test (also called “non-directional test” or “two-tailed test”)- determines if the sample mean is from a population with mean that differs significantly

from the hypothesized mean.- determines if two groups significantly differ.

*The type of test is determined by analyzing the problem, specifically on the last question. Or is depending on the hypothesis.

Level of significance (α)- a pre-determined probability value that a researcher wants in committing a maximum error upon stating

the conclusion- sets the area of rejection where the conclusion will be based- usually, values are α = 0.10, α = 0.05, or α = 0.01- α = 1 – confidence interval.

STEPS IN HYPOTHESIS TESTING:1. Analyze the problem to determine the null and alternative hypotheses.2. Set the level of significance.3. Determine the test statistic to be used (z-test, t-test, etc)4. Compute for the statistic value.5. Find the critical value and set the area of rejection.6. If the obtained statistic value is in the critical region (that is beyond the critical value(s)),

“Reject Ho”. Otherwise, “Fail to reject Ho”.

Table of critical values for z-tests: *Based on the areas under the normal curve.

Rejection of Ho means that a researcher has found sufficient evidence against the null hypothesis.

Non-rejection of Ho means that a researcher has not found sufficient evidence against the null hypothesis and thus, there is a reason to believe that the null hypothesis is true.

Given the following research questions involving tests of single mean, determine whether it leads to a one or two-tailed test. Then state Ho and Ha.

a) Is the mean IQ of Thomasians significantly above 120?� One-tailed test � Two-tailed test

Ho: _________________________________________________________________

Ha: _________________________________________________________________

b) Do the skin cancer patients’ have a mean body temperature of less than 37ºC two hours after applying fluorouracil?� One-tailed test � Two-tailed test

Ho: _________________________________________________________________

Ha: _________________________________________________________________

c) Does biliary atresia occur to Filipino children at a mean of 3 years old?� One-tailed test � Two-tailed test

Ho: _________________________________________________________________

Ha: _________________________________________________________________

TESTS FOR SINGLE POPULATION MEAN

- determines if a population mean is equal to the hypothesized mean

or *t-test for single mean uses degrees of freedom (df) equal to n – 1.

1. Suppose that a researcher has conducted a study to determine whether the weight of 12-year old boys inhis town is 80 lbs. He collected 50 samples with a mean weight of 77 lbs. If the population standard deviation known to be 9 lbs, can he conclude that the mean weight of the 12-year old boys in his town is not equal to 80 lbs? Test at α = 0.05.

Hypotheses:

Ho: ____________________________________________________________________________

Ha:______________________________________________________________________________

Test Statistic: Critical Value: ________________________

Decision: _______________

Conclusion: ______________________________________________________________________

2. The mean weight of the sample of 40 persons from the Honolulu Heart Study is 64 kg. If the ideal weight is known to be 60 kg, is the group significantly overweight? Assume ߪ = 15 kg and α = 0.05.

Hypotheses:

Ho: ____________________________________________________________________________

Ha:______________________________________________________________________________

Test Statistic: Critical Value: ________________________ p-value: ___________

Decision: _______________

Conclusion: ______________________________________________________________________

3. From a sample of 25 patients, the mean diastolic blood pressure in Honolulu Hospital is 72.73 mmHg with a standard deviation of 6.97 mmHg. At α = 0.05, test whether the mean blood pressure of this group is significantly greater than 70.Filename: ex11.savHypotheses:

Ho: ____________________________________________________________________________

Ha:______________________________________________________________________________

Test Statistic: Critical Value: ________________________ p-value: ___________

Decision: _______________

Conclusion: ______________________________________________________________________

4. A quality control specialist of a pharmaceutical company wants to determine if all the active ingredient of the capsules being manufactured has a mean mass of 500 mg. A sample of 15 capsules were taken and found a mean of 504.31 mg with a standard deviation of 9.85 mg. Can the quality control specialist conclude that the mean mass of all the capsules being manufactured is not 500 mg? Test at α = 0.05.Filename: ex12.sav

Hypotheses:

Ho: ____________________________________________________________________________

Ha:______________________________________________________________________________

Test Statistic: Critical Value: ________________________ p-value: ___________

Decision: _______________

Conclusion: ______________________________________________________________________

Homework – Inference in Single Mean

Names: ____________________________________ Section: __________ Score: _______ / 15 ____________________________________

1. A bottle of insect repellant is labeled Net contents: 16 Fl. oz. The Bureau of Standards randomly selected 15 bottles from as many stores in a city and measured the actual contents. The results (in ounces) show that the mean is 15.8 and standard deviation is 0.29. Does the data provide sufficient evidence that mean Net concent (in Fl oz.) of the insect repellant is not 16 Fl.oz? Test at α=0.05.Hypotheses:

Ho: _____________________________________________________________________________________

Ha: _____________________________________________________________________________________

Test Statistic: Critical Value: ________________________

Decision: _______________

Conclusion: _______________________________________________________________________________

2. A manufacturer of medicine claimed that the percentage of absorbance of their paracetamol is 42.29. A researcher has drawn 50 random samples and found out that the mean absorbance of this paracetamol is 39.55 with a standard deviation of 0.87. Is there a sufficient evidence to conclude that the mean absorbance of this paracetamol is below 42.29? Test at α=0.05.Hypotheses:

Ho: _____________________________________________________________________________________

Ha: _____________________________________________________________________________________

Test Statistic: Critical Value: ________________________

Decision: _______________

Conclusion: _______________________________________________________________________________

3. An experiment was done to test whether the cholesterol density of the residents in a certain barangay is 200mg/dL. Twenty five residents were randomly selected and have found to have a mean cholesterol density of 257 mg/dL. Assuming that the cholesterol density’s standard deviation of all the residents is 45 mg/dL, is there a sufficient evidence to conclude that the mean cholesterol density of these residents is above 200 mg/dL at α=0.05.Hypotheses:

Ho: _____________________________________________________________________________________

Ha: _____________________________________________________________________________________

Test Statistic: Critical Value: ________________________

Decision: _______________

Conclusion: _______________________________________________________________________________