Case Study 473 1. If there have been no temperature trends over the years, then it will be equally likely for the record high or the record low to be more recent. If there has been a warming trend, it might be more likely for the record high to be more recent. Let p be the probability that the record high occurred more recently than the record low. Use the sample proportion of dates where the high occurred more recently to test H0 : p = 0.5 versus H1: p > 0.5. What do you conclude? 2. The following table presents the records for every day in June. The data show that it is common for records to be set on two or more consecutive days in the same year. This is due to hot spells and cold spells in the weather. For example, five consecutive record highs, from June 2 through June 6, occurred in 1925. Explain why using data for every day may violate the assumption, used in Exercise 1, that the data are a simple random sample. Date High Year Low Year Date High Year Low Year Date High Year Low Year Jun 1 98 2011 45 1938∗ Jun 11 101 1911 45 1913 Jun 21 99 2012 51 1940 2 97 1925∗ 43 1897 12 95 2002∗ 50 1907∗ 22 101 1988 51 1992∗ 3 99 1925 45 1910 13 96 1954 51 1887 23 98 1988∗ 51 1918 4 99 1925 46 1929∗ 14 98 1994 49 1933 24 100 2010 46 1902 5 100 1925 48 1926 15 101 1994 47 1933 25 100 1997 53 1902 6 97 1925 46 1945∗ 16 99 1994 50 1917 26 101 1952 56 1979 7 98 2008∗ 47 1894 17 95 1991∗ 50 1926 27 99 2010 57 1927∗ 8 99 2011 49 1977∗ 18 97 1944 51 1965∗ 28 100 1969 54 1927 9 102 2011∗ 45 1913∗ 19 99 1994 51 1909 29 104 2012 54 1888 10 100 1964 46 1913 20 99 1931 54 1926∗ 30 100 1959 50 1919 ∗Indicates that the record occurred more than once; only the most recent year is given. 3. We will perform another test to determine whether record highs are more likely to have occurred recently. If a record high is equally likely to occur in any year of observation, the mean year in which a record is observed would occur at the midpoint of the observation period, which is (1871 + 2013)∕2 = 1942. Use the data in Table 9.4 to test the hypothesis that the mean year in which a record high occurred is 1942 against the alternative that it is greater. What do you conclude? 4. For some records, marked with a *, the record temperature occurred more than once. In these cases, only the most recent year is listed. Explain how this might cause the mean to be greater than the midpoint of 1942, even if records are equally likely to occur in any year. 5. Using the data in Table 9.4, drop the dates in which the record high occurred more than once, and test the hypothesis in Exercise 3 again. Does your conclusion change? No 6. Perform a hypothesis test on the record lows, after dropping dates on which the record low occurred more than once, in which the alternate hypothesis is that the mean year is less than 1942. What do you conclude? 7. Using the analyses you have performed, write a summary of your findings. Describe how strong you believe the evidence to be that record highs have tended to occur more recently than record lows.
navidi_monk_elementary_statistics_2e_ch7-9
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