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Chapter 9 Elementary Statistics (Hypothesis Testing/One sample Analysis) Statistical hypothesis: A statistical hypothesis is a conjecture (proposition) about a population parameter. This conjecture may or may not be true. Null hypothesis: The null hypothesis denoted by H 0 is a statistical hypothesis that states that there is no difference between a parameter and a specific value (One-sample analysis), or there is no difference between two parameters (Two-sample analysis). Alternative hypothesis: The alternative hypothesis denoted by H1 is a statistical hypothesis that states the existence of a difference between a parameter and a specific value (One-sample analysis), or states that there is a difference between two parameters. (Two-sample analysis) Possible scenarios: Hypothesis testing about the population mean H 0 : µ = µ0 H1 : µ ≠ µ 0 (Two-tail test) µ H 0 : µ ≤ µ0 H1 : µ > µ0 (Right-tail test) H 0 : µ ≥ µ0 H1 : µ < µ 0 (Left-tail test) Hypothesis testing about the population proportion P H 0 : P = P0 H1 : P ≠ P0 (Two-tail test) H 0 : P ≤ P0 H1 : P > P0 (Right-tail test) H 0 : P ≥ P0 H1 : P < P0 (Left-tail test) The hypothesis-testing situation can be linked to a jury trial. In a jury trial, there are four possible outcomes. The defendant is either guilty or innocent, and he or she will be convicted or acquitted. 1 Type I Error: This error occurs if one rejects the null hypothesis when it is true. Type II Error: This error occurs if one does not reject the null hypothesis when it is false. The level of significance: The level of significance is the maximum probability of committing a type I error. This probability is denoted by α (Usually α ≤ 0.1 ) . [Note that the probability of making type II error is denoted by β ] Methods of analysis: a) The traditional method b) The p-value method c) The confidence-interval method Examples: For each conjecture, state the null and alternative hypotheses using the mathematical symbols. a. The average age of community college students is 24.6 years. H 0 : ---------------------------H1 : ---------------------------b. The average age of attorneys is greater than 25.4 years. H 0 : ---------------------------H1 : ---------------------------- c. The average score of 50 high school basketball games is less than 88. H 0 : ---------------------------H1 : ---------------------------- d. The average cost of DVD player is $79.95. H 0 : ---------------------------H1 : ---------------------------- 2 The Traditional Method: (we will skip this method) The P-value (Probability value): Assuming that the null hypothesis is true, the P-value can be defined as the probability of obtaining a sample statistics such as x , p̂ and s with a value as extreme or more extreme than the one determined from the sample data. The P-value is the smallest significance level at which the null hypothesis is rejected. Using the Pvalue method, we reject the null hypothesis if p − value < α and we do not reject the null hypothesis if p − value ≥ α . I) Hypothesis testing about µ and σ is known: [Either n ≥ 30 or population is normal with n < 30 ] This case is not a realistic situation and we will skip it. II) Hypothesis testing about µ and σ is not known: [Either n ≥ 30 or population is normal with n < 30 ] 1. Cushman and Wakefield reported that the average annual rent for office space in Pomona, Ca was $17.63 per square foot. A real estate agent selected a random sample of 15 rental properties (offices) and found the mean rent was $18.72 per square foot, and the standard deviation was $3.64. At α = 0.05 , test the claim that there is no difference in the rents. (Use the P-value method) a. H 0 : − − − − − − H1 : − − − − − − c. P α b. The P-value value: -------------------------------- : ------------------------------------ d. The initial conclusion: ---------------------------- e. The final conclusion:-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------TI: STAT → TESTS → T Test → Stats → Type-in the values for µ 0 , x , S x , n , and choose the proper sign for µ → Calculate 3 2. A researcher estimates that the average height of the buildings of 30 or more stories in a large city is at least 700 feet. A random sample of 10 buildings is selected, and the heights in feet are shown. At α = 0.025 , is there enough evidence to reject the claim? (Use the P-value method) 485 a. 511 841 725 615 H 0 : − − − − − − H1 : − − − − − − c. P ------ 520 535 635 616 582 b. The P-value value: -------------------------------- α d. The initial conclusion: ---------------------------- e. The final conclusion:------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------TI: STAT → TESTS → T Test → Data → Type-in the values for µ 0 , and choose the proper sign for µ → Calculate III) Hypothesis testing about population Proportion P: Large Samples: [i.e. n pˆ > 5 and n qˆ > 5 , where qˆ = 1 − pˆ ] 3. It has been reported that 40% of the adult population participates in computer hobbies during their leisure time. A random sample of 180 adults found that 65 engaged in computer hobbies. At α = 0.01 , is there sufficient evidence to conclude the proportion differs from 40%? (Use the P-value method) a. H 0 : − − − − − − H1 : − − − − − − c. P ------ α b. The P-value value:-------------------------------- d. The initial conclusion: ---------------------------- e. The final conclusion:------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------TI: STAT → TESTS → 1 Prop Z Test → Enter → Type-in the values for p0 , x, n, and choose the proper sign for p0 → Calculate 4. The percentage of physicians who are women is 27.9%. In a survey of physicians employed by a large university health system, 45 of 120 randomly selected physicians were women. Is there sufficient evidence at the 0.05 level of significance to conclude that the proportion of women physicians at the university health system exceeds 27.9%? (Use the P-value method) a. H 0 : − − − − − − H1 : − − − − − − c. P ------ α b. The P-value value: -------------------------------- d. The initial conclusion: ---------------------------- e. The final conclusion:------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------TI: STAT → TESTS → 1 Prop Z Test → Enter → Type-in the values for p0 ,x, n, and choose the proper sign for p0 → Calculate 4