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HYPOTHESIS TESTING EXAMPLES - 3 1. Suppose a well-known doctor believes that at least 75% of women wear shoes that are too small. A study of 356 women by an orthopedic group found 313 women that wore shoes that were too small. Using a 0.03 level of significance, is the doctor’s belief supported by this study? Ho: π = 0.75 Ha: π > 0.75 Z p (1 ) n 0.88 0.75 5.66 0.75(1 0.75) 356 Reject Ho if Z > 1.89 Reject Ho There is sufficient evidence to conclude the proportion of women who wear shoes that are too small is greater than 0.75, α = 0.03. 2. Suppose, according to a 1990 demographic report, the average U. S. household spends $90 per day. Suppose you recently took a random sample of 30 households in Huntsville and the results revealed a mean of $84.50 and a standard deviation of $14.50. Using a 0.05 level of significance, can it be concluded that the average amount spent per day by U.S. households has decreased? Ho: μ = 90 Ha: μ < 90 t x 84.50 90 -2.078 s 14.50 n 30 Reject Ho if t < -1.699 Reject Ho There is sufficient evidence to conclude the average amount spent per day by U.S. households has decreased, α = 0.05. 3. Suppose the mean salary for full professors in the United States is believed to be $61,650. A sample of 36 full professors revealed a mean salary of $69,800. Assuming the standard deviation is $5,000, can it be concluded that the average salary has increased using a 0.02 level of significance? Ho: μ = 61,650 Ha: μ > 61,650 Z x n 69,800 61,650 9.78 5,000 36 Reject Ho if Z > 2.06 Reject Ho There is sufficient evidence to conclude the average salary has increased, α = 0.02. 4. Historically, evening long-distance calls from a particular city have averaged 15.2 minutes per call. In a random sample of 35 calls, the sample mean time was 14.3 minutes with a standard deviation of 5 minutes. Using a 0.05 level of significance, is there sufficient evidence to conclude that the average evening long-distance call has decreased? Ho: μ = 15.2 Ha: μ < 15.2 t x 14.3 15.2 -1.065 s 5 n 35 Reject Ho if t < -1.697 Fail to Reject Ho There is insufficient evidence to conclude the average evening long-distance call has decreased, α = 0.05. 5. Suppose a production line operates with a mean filling weight of 16 ounces per container. Since over- or under-filling can be dangerous, a quality control inspector samples 30 items to determine whether or not the filling weight has to be adjusted. The sample revealed a mean of 16.32 ounces. From past data, the standard deviation is known to be 0.8 ounces. Using a 0.10 level of significance, can it be concluded that the process is out of control (not equal to 16 ounces)? Ho: μ = 16 Ha: μ ≠ 16 Z x n 16.32 16 2.191 0.8 30 Reject Ho if Z < -1.65 OR Z > 1.65 Reject Ho There is sufficient evidence to conclude the process is out of control, α = 0.10. 6. Suppose in 1991, the proportion of individuals in this country trying to follow guidelines on eating right was .44. In 1995, a survey of 1,000 individuals revealed that 497 were trying to follow guidelines on eating right. Has the proportion of individuals in this country trying to follow guidelines on eating right increased? Use a 0.08 level of significance. Ho: π = 0.44 Ha: π > 0.44 Z p (1 ) n 0.497 0.44 3.63 0.44(1 0.44) 1,000 Reject Ho if Z > 1.41 Reject Ho There is sufficient evidence to conclude the proportion of individuals in this country trying to follow guidelines on eating right has increased, α = 0.08.