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STATISTICS FOR BUSINESS. Chapter 8. Hypothesis testing for a single population. A nuclear power plant adjacent to a residential area. STATISTICS FOR BUSINESS (Hypothesis testing for a single population). A nuclear power plant adjacent to a residential area.

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1. STATISTICS FOR BUSINESS Chapter 8. Hypothesis testing for a single population A nuclear power plant adjacent to a residential area

2. STATISTICS FOR BUSINESS (Hypothesis testing for a single population) A nuclear power plant adjacent to a residential area An hypothesis is giving an opinion or making a decision without objective information Consider: Government announces "Radiation levels around a nuclear power plant are well below levels considered harmful". Two local residents die of leukemia Does this event make us conclude that the government is giving wrong information? It is wrong to accept, or reject, a hypothesis about a population parameter simply by intuition. One needs to decide objectively on the basis of measured sample information.

3. STATISTICS FOR BUSINESS (Hypothesis testing for a single population) A nuclear power plant adjacent to a residential area Procedure for hypothesis testing • Select a random sample • Measure the appropriate statistic - the mean or proportion • Decide on the desired level of significance: (Say 5%) • Determine if the statistic falls within an appropriate region of acceptance • Accept the hypothesis if the statistic falls into the acceptance region. • Otherwise, reject it Even if a sample statistic does fall in the area of acceptance, it does not prove that the null hypothesis, Ho, is true. There is simply no statistical evidence to reject it.,

4. STATISTICS FOR BUSINESS (Hypothesis testing for a single population) A nuclear power plant adjacent to a residential area Nomenclature in hypothesis testing • Hypothesis: • In a certain country, average age of population is 35 within a given significance level • Written as: Ho:µx= 35 • Null hypothesis is that population mean is equal to 35 • Alternative hypothesis: • Population mean is not equal to 35. • That is the mean or average age is significantly different from 35 • Written as: H1:µx35 • Whenever the null hypothesis is rejected, accepted conclusion is the alternative hypothesis • Binomial either “accept” or “reject”

5. STATISTICS FOR BUSINESS (Hypothesis testing for a single population) A nuclear power plant adjacent to a residential area Concept of significance • Exam grades - Case 1 • John has an A in the course on Business Statistic: Susan has an A • Is the difference significant? NO • Exam grades - Case 2 • Sarah has an A in the course on Business Statistics: Derek has a C- • Is the difference significant? YES • Ages - Case 1 • Joan, Susan, and Mike are in the same class at university. • Is there a significance difference in their age? PROBABLY NOT • Ages- Case 2 • Angela is the granddaughter of Kenneth • Is there a significant difference in their ages? YES • Automobile prices - Case 1 • Erin has just bought a new red Austin Mini automobile. Peter has just bought the same model, but green. • Is there a significance difference in their purchase price? PROBABLY NOT • Automobile prices - Case 2 • Pauline has just bought a new Austin Mini automobile. Jeffrey has just bought a Porsche. • Is there a significant difference in their purchase price? YES

6. STATISTICS FOR BUSINESS (Hypothesis testing for a single population) A nuclear power plant adjacent to a residential area Two-tailed hypothesis test Question asked: "Is there evidence of a difference?" Null hypothesis: Average age of a certain group is 35 years: Ho:µx= 35 Alternative: Is there evidence that average age the group is different than 35 years: H1:µx ≠ 35 At 10% significance, there is 5% in each tail If sample means falls within the non shaded area, accept the null hypothesis Reject the null hypothesis if sample mean falls in either of the shaded regions

7. STATISTICS FOR BUSINESS (Hypothesis testing for a single population) A nuclear power plant adjacent to a residential area One-tailed right hand hypothesis test Question asked: "Is there evidence of a being greater than?" Null hypothesis is that average age of a certain group is not greater than 35 years: Ho:µx ≤ 35 Alternative: Is there evidence that average age the group is greater than 35 years: H1:µx > 35 At 10% significance, there is 10% in right hand tail If sample means falls within the non shaded area, accept the null hypothesis Reject the null hypothesis if sample mean falls in the shaded region

8. STATISTICS FOR BUSINESS (Hypothesis testing for a single population) A nuclear power plant adjacent to a residential area One-tailed left hand hypothesis test Question asked: "Is there evidence of being less than?" Null hypothesis: Average age of a certain group is not less than 35 years: Ho:µx ≥ 35 Alternative hypothesis: Is there evidence that average age the group is less than 35 years: H1:µx < 35 At 10% significance, there is 10% in left tail If sample means falls within the non shaded area, accept the null hypothesis Reject the null hypothesis if sample mean falls in the shaded region

9. STATISTICS FOR BUSINESS (Hypothesis testing for a single population) A nuclear power plant adjacent to a residential area Selecting a significance level • Higher the significance level for testing the hypothesis, greater • is probability of rejecting a null hypothesis when it is true. • However, in this case we would rarely accept a null hypothesis • when it is not true. 99% 90% Significance level of 1% (0.5% in each tail) Significance level of 10% (5% in each tail) 50% Significance level Is total area in the tails Significance level of 50% (25% in each tail)

10. STATISTICS FOR BUSINESS (Hypothesis testing for a single population) A nuclear power plant adjacent to a residential area Hypothesis testing for the mean with sample sizes greater than 30. Using Normal distribution observed mean hypothesized mean Test statistic is: z can be + or - Population standard deviation sxis known. Large samples • Numerator measures how far the observed mean is from hypothesized mean. • Denominator is standard error • z represents how many standard errors observed mean is from hypothesized mean

11. STATISTICS FOR BUSINESS (Hypothesis testing for a single population) A nuclear power plant adjacent to a residential area Hypothesis testing for the mean with sample sizes less than 30. Using Student-t distribution hypothesized mean observed mean Test statistic is: t can be + or - Population standard deviation sxis unknown. Only standard deviation available is sample standard deviation, s. Small samples • If population is assumed to be normally distributed, sampling distribution of mean • will follow a t distribution with (n - 1) degrees of freedom. n is the sample size less than 30

12. STATISTICS FOR BUSINESS (Hypothesis testing for a single population) A nuclear power plant adjacent to a residential area p-value approach to hypothesis testing • The p-value (probability value) is the observed level of significance • It is the smallest level at which H0 can be rejected for a given set of data. • The p-value answers the question, “If H0is true, what is the probability of obtaining x-bar or ps, this far or more from H0?” • If the p-value from sample is greater than, or equal to a the null hypothesis should be accepted • If the p-value is less than a the null hypothesis should be rejected

13. STATISTICS FOR BUSINESS (Hypothesis testing for a single population) A nuclear power plant adjacent to a residential area Comparing sample value with critical limits • From the established significant level determine the limits. Either Normal z, or Student t • Established whether the sample value lies within these limits ; • If it does accept the null hypothesis. If not reject the null hypothesis.

14. STATISTICS FOR BUSINESS (Hypothesis testing for a single population) A nuclear power plant adjacent to a residential area Types of errors TYPE I Error Rejecting a null hypothesis when it is in fact true: Probability of a Type 1 error is alpha Alpha is the level of significance TYPE II Error Accepting a null hypothesis, when it is in fact false Probability of a Type II error is Beta • A Type I error involves time & cost of reworking a batch of chemicals that should have been accepted • A Type II error, means taking a chance than an entire group of users of the chemical will be poisoned • Management would prefer a Type I error. Potential risk is lower • Making a Type I error involves shutting down and modifying an entire assembly line at a work center. • Making a Type II error, involves less expensive warranty repairs at the dealers • Management would prefer a Type II error. Less costly! (Ethics?) • Under Anglo Saxon criminal law an individual is considered innocent of a certain crime. • Guilt must be proven. • Preferable to commit a Type II error (Accepting a null hypothesis when it is false) and let a guilty person • go free, rather than perhaps sentence an innocent person for a crime they did not commit.

15. STATISTICS FOR BUSINESS (Hypothesis testing for a single population) A nuclear power plant adjacent to a residential area Hypothesis testing of proportions • Binomial is correct distribution • Success • Failure If n.p and n.q are both 5: Normal distribution can be used to approximate the sampling distribution; Test value of proportion, p-bar Hypothesized proportion As for the mean there can be a two tail test, or a one tail test

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