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Statistics β Hypothesis Testing (Ch 9) β Review. KEY You should complete this review, with work shown, to be eligible to test. You may use a 3x5 card of notes on the test. All probability answers must be either a completely reduced fraction, a decimal rounded to 4 places or a percent rounded to 2 places. Unless instructed otherwise, round any other response to 4 decimal places. 1. State the requirements that must be satisfied in order to test a claim about a population mean when the population standard deviation is known. What type of distribution is used? Normal distribution (z-scores) 1. Sample is randomly selected. 2. Either π β₯ ππ or the sample is normally distributed with no outliers. 2. State the requirements that must be satisfied in order to test a claim about a population mean when the population standard deviation is not known. What type of distribution is used? t-distribution (t-scores) 1. Sample is randomly selected. 2. Either π β₯ ππ or the sample is normally distributed with no outliers. 3. State the requirements that must be satisfied in order to test a claim about a population proportion. What type of distribution is used? Normal distribution (z-scores) 1. Sample is randomly selected. 2. Sample must be no more than 5% of the population. 3. πππ (π β ππ ) β₯ ππ 4. State the requirements that must be satisfied in order to test a claim about a population standard deviation. What type of distribution is used? ππ distribution (ππ -scores) 5. Explain what πΌ represents. The probability of making a Type I error. The probability that an innocent man is convicted. H o is rejected when it is true. 6. Explain what π½ represents. The probability of making a Type II error. The probability that a guilty man is goes free. Ho is not rejected when it should be. 7. Explain what the power of the test tells us about the hypothesis test. This is the probability that when the Null (Ho) is false we reject the Null. When the power of the test is close to 1, we are not very likely to make a Type II error (let the guilty man go free). When the power of the test is close to 0, we are likely to make a Type II error and not likely to make a Type I error. For 8 β 11, test the hypothesis using both the classical and the P-value approaches. Clearly state your conclusion. 8. A certain brand of apple juice is supposed to have 64 ounces of juice. Because the filling machine is not precise, the exact amount of juice varies from bottle to bottle. The quality control manager claims that the mean amount of juice in each bottle is not 64 ounces. She randomly samples 22 bottles of juice, measures the content, and obtains a sample mean of 63.96 ounces. Assuming that ο³ = 0.06 and the distribution is normal, test her claim at the ο‘ = 0.1 level of significance. Two tailed test. Test statistic: π = βπ. ππ, Critical Values: π = ±π. ππ, P-value = 0.0018. Reject the Null There is sufficient evidence to support the claim that the average amount of juice in each bottle is not 64 ounces. 9. In order to test H0: ο = 40 versus H1: ο > 40, a simple random sample of size n = 25 is obtained from a population that is known to be normally distributed. A sample mean is found to be 42.3 and a sample standard deviation of 4.3. Test the researcherβs claim the ο‘ = 0.1 level of significance. Right tailed test. Test statistic: π = π. ππ, Critical Value: π = π. ππ, P-value = 0.0066. Reject the Null There is sufficient evidence to support the claim that the mean is larger than 40. 10. In a recent survey, 395 out of 1012 adults aged 18 years or older said they had a gun in the house. In 1990, 47% of households had a gun. Is there significant evidence to support the claim that the proportion of households that have a gun has decreased since 1990 at the ο‘ = 0.05 level of significance? Μ = βπ. πππ, Critical Value: π = βπ. ππ, P-value = 0.00000019. Reject the Null Left tailed test. Test statistic: π There is sufficient evidence to support the claim that gun ownership has declined since 1990. 11. A researcher is trying to determine if the standard deviation for the amount of acidity in rain has changed from 0.2. He obtains 19 samples and finds that the standard deviation is 0.1708. Test the researcherβs claim at the ο‘ = 0.05 level of significance. Assume that the population is normal. Two tailed test. Test statistics: ππ = ππ. πππ, Critical Value: ππ = π. πππ, ππ = ππ. πππ, P-value = 0.4321. Do not reject Ho There is not sufficient evidence to support the claim the standard deviation is different from 0.2. For 12 β 13, find the probability of making a Type II error and find the power of the test. 12. A researcher maintains that the average age of a woman before she has her first child is greater than the 1990 average age of 26.4 years. She obtains a simple random sample of 40 women who recently gave birth to their first child. Assume that the population standard deviation is 6.4 years. If the researcher decides to test this hypothesis at the ο‘ = 0.05 level of significance, the true population mean is 26.9 years. Right tailed. Critical Value: π = π. ππ, in context 28.064, π· = π. ππππ, power of the test = 0.1249 a. In context, what does π½ tell us about this test? We are very likely to conclude the average age of a woman is 26.4 years old before having her first child, when in fact she is older than 26.4 years. b. In context, what does the power of the test tell us about this test? We are not likely to conclude that the average age of a woman is more than 26.4 years when she is in fact 26.4 years old. 13. A school administrator claims that students whose first language learned is not English score different on the verbal portion of the SAT exam than students whose first language is English. The mean SAT verbal score of students whose first language is English is 516. Suppose a simple random sample of 20 students whose first language learned was not English. SAT verbal scores are normally distributed with a population standard deviation of 109. If the researcher decides to test this hypothesis at the ο‘ = 0.05 level of significance, compute the probability of making a Type II error if the true population mean is 505. Two tailed. Critical Values: π = ±π. ππ, in context 468.23 and 563.770, π· = π. ππππ, power of the test = 0.0736 a. In context, what does π½ tell us about this test? We are very likely to conclude that the average SAT verbal score is 516 when in fact it is not. b. In context, what does the power of the test tell us about this test? We are not very likely to conclude that the average SAT verbal score is not 516 when in fact it is.