Prob. & Stat. Lecture10 - one-/two-sample tests of hypotheses ([email protected]) 10-1 1036: Probability & Statistics 1036: Probability & 1036: Probability & Statistics Statistics Lecture 10 Lecture 10 – – One One - - and Two and Two - - Sample Sample Tests of Hypotheses Tests of Hypotheses
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• The acceptance of a hypothesis merely implies that the data do not give sufficient evidence to refute it.
• Rejection means that there is a small probability of obtaining the sample information observed when the hypothesis is true.
• Example: for the conjecture of the fraction defective p = 0.10, a sample of 100 revealing 20 defective items is certainly evidence of rejection– Since the probability of obtaining 20 defectives is
approximately 0.002• The firm conclusion is established by the data analyst when
Hypothesis Testing with a Continuous Random Variable
• Consider the null hypothesis that the average weight of male students in a certain college is 68 kilograms against the alternative hypothesis that it is unequal to 68.– H0 : µ = 68; H1: µ ≠ 68
Properties of a Test Hypothesis• The type I error and type II error are related. A decrease
in the probability of one generally results in an increase in the probability of the other
• The size of the critical region, and therefore the probability of committing a type I error, can always be reduced by adjusting the critical value(s).
• An increase in the sample size n will reduce α and βsimultaneously
• If the null hypothesis is false, β is a maximum when the true value of a parameter approaches the hypothesized value. The greater the distance between the true value and the hypothesized value, the smallerβ will be
How are H0 and H1 Chosen?• Example 10.1: A manufacturer of a certain brand of rice cereal
claims that the average saturated fat content does not exceed 1.5 grams. State the null and alternative hypotheses to be used in testing the claim and determine where the critical region is located.– The claim should be rejected only if µ is greater than 1.5– One-tailed test– H0 : µ = 1.5; H1: µ > 1.5
• Example 10.2: A real estate agent claims that 60% of all privateresidences being built today are 3-bedroom homes. State the null and alternative hypotheses to be used in testing the claim and determine the location of the critical region.– The higher or lower test statistic than 0.6 would reject the claim– Two-tailed test– H0 : p = 0.6; H1: p ≠ 0.6
P-Values for Decision Making• It had become customary to choose an α of 0.05
or 0.01 and select the critical region accordingly. (to control the type I error)
• However, this approach does not account for values of test statistics that are close to the critical region
• A P-value is the lowest level of significance at which the observed value of the test statistic is significant– no fixed α is determined – The conclusion is made on the basis of p-value in
P-value ApproachSignificant testing approach– State null and alternative hypotheses.– Choose an appropriate test statistic.– Compute P-value based on computed value of
test statistic. – Use judgment based on P-value and knowledge
Example 10.3• A random sample of 100 recorded deaths in the United
States during the past year showed an average life span of 71.8 years. Assuming a population standard deviation of 8.9 years, does this seem to indicate that the mean life span today is greater than 70 years? Use a 0.05 level of significance.
Example 10.4• A manufacturer of sports equipment has developed a new synthetic
fishing line that he claims has a mean breaking strength of 8 kilograms with a standard deviation of 0.5 kilogram. Test the hypothesis that μ= 8 kilograms against the alternative that μ≠ 8 kilograms if a random sample of 50 lines is tested and found to have a mean breaking strength of 7.8 kilograms. Use a 0.01 level of significance.
• For the case of a single population with mean µ and variance σ2 known, both hypothesis testing and confidence interval estimation are based on the R.V.
• We have (1-α)×100% confidence interval on µ• The testing of H0: µ=µ0 against H0: µ≠µ0 at a significance
level α and rejecting H0 if µ0 is not inside the confidence interval