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Data Analysis For
Environmental Applications
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Large Sample Two-sided Hypothesis Tests for a Single Population
The NOAA National Climatic Data Center (NCDC) provides climatological records for many
cities in the US. For example, you can find a record of annual precipitation totals at Boston at:
Boston_precip.txt
For this exercise, you should use the NOAA data set to test the following hypothesis about the
mean annual rainfall at Boston:
H0: a = E[annual rainfall]= 100cm
Derive a test statistic for three cases, using the first 10 years, first 40 years, and all 90 years of
data, respectively. For each case, determine whether or not you can reject H0 at the 5%
significance level (i.e. for =0.5). Make a “large sample” assumption and use a two-sided
test. Also report the p value for the test.
Problem 2: Large Sample Two-sided Hypothesis Tests for Two Populations
The Massachusetts Water Resources Authority (MWRA) conducts an ongoing program of
monitoring bacterial levels in Boston Harbor and its tributaries. An example of some of the data
collected is provided in the file MWRA1.txt . This file lists the coliform count C (reported in
organisms per 100 ml) from the indicated stations and dates. The stations can be grouped as
follows:
Outer harbor (control): 104, 105, 118, 81, 82 Inner harbor
(affected): 19,20
Check the null hypothesis that the means of the inner and outer harbor coliform levels are the
same. Work with the transformed concentration CT = ln (C+1). Report the p level for a large
sample two-sided test.
Data Analysis For Environmental Applications Uncertainty in Engineering

Some relevant MATLAB functions: normcdf,norminv,normplot, mean,var
Problem 3: Small Sample Two-sided Hypothesis Test for a Single Population, based on Stochastic
Simulation
Suppose that you wish to test the hypothesis that the standard deviation of a lognormal population is
equal to its mean (i.e. HO:SD(x) - E(x) = 0). You are given the following small sample:
[x1, x2, x3, x4, x5] = [1.76 0.51 3.08 3.62 0.86]
Suggest an appropriate standardized small sample test statistic and use stochastic simulation to derive an
approximate CDF for this statistic. Then determine the p value for a double-sided test of the hypothesis.
Problem 4: Small Sample Two-sided Hypothesis Test for a Single Population, based on the Chi-
squared Statistic
Repeat the above problem, assuming that the population is normally distributed so that the Chi-
squared statistic and CDF can be used. In this case, use the following small sample:

**Note:** This dataset is not for commercial distribution.
Total Precipitation in Centimeters
1900 111.89
1901 123.75
1902 86.19
1903 106.59
1904 100.68
1905 81.48
1906 103.37
1907 95.40
1908 76.37
1909 103.30
1910 71.98
1911 90.88
1912 87.77
1913 96.81
1914 87.16
1915 98.45
1916 94.90
1917 99.02
1918 87.36
1919 108.46
1920 116.37
1921 109.12
1922 104.48
1923 88.62
1924 88.70
1925 104.53
1926 101.67
1927 104.28

1928 95.51
1929 94.46
1930 89.63
1931 112.19
1932 105.79
1933 120.93
1934 86.74
1935 83.67
1936 117.61
1937 110.91
1938 128.38
1939 82.45
1940 91.56
1941 78.40
1942 108.10
1943 81.90
1944 94.20
1945 120.10
1946 96.70
1947 96.20
1948 113.50
1949 80.00
1950 83.20
1951 119.30
1952 103.20
1953 146.60
1954 158.30
1955 143.40
1956 120.60

1957 86.10
1958 156.40
1959 130.59
1960 113.00
1961 121.50
1962 109.80
1963 88.70
1964 92.60
1965 60.30
1966 91.60
1967 121.20
1968 107.40
1969 121.40
1970 106.45
1971 90.62
1972 134.90
1973 108.58
1974 102.21
1975 116.30
1976 93.27
1977 112.19
1978 95.60
1979 112.20
1980 74.66
1981 90.69
1982 113.32
1983 136.15
1984 127.61
1985 92.93
1986 112.62
1987 115.51
1988 88.33
1989 107.75

STAT_ID VALUE DATEONLY
104 0 10/7/92
104 1450 11/18/91
104 1760 11/19/91
104 1200 11/20/91
104 10200 11/21/91
104 5 5/20/92
104 0 6/4/92
104 5 10/8/92
104 10 10/9/92
104 5 10/14/92
104 10 10/15/92
104 0 5/21/92
105 5 11/20/91
105 60 11/19/91
105 200 11/21/91
105 0 5/20/92
105 0 5/21/92
105 0 6/4/92
105 5 10/7/92
105 5 10/8/92
105 15 10/9/92
105 30 10/14/92
105 15 10/15/92
105 45 11/18/91
118 20 8/6/94
118 0 8/9/93
118 0 8/11/93
118 0 8/12/93
118 0 8/19/92
118 5 8/17/92
118 0 8/15/92
118 0 8/14/92
118 5 8/13/92
118 0 8/12/92
118 0 8/2/94

118 35 8/3/94
118 0 7/25/93
118 0 8/5/94
118 0 8/13/93
118 0 8/9/94
118 0 8/11/94
118 0 8/12/94
118 0 8/16/94
118 0 8/19/94
118 55 8/20/94
118 0 8/10/92
118 0 8/8/92
118 0 8/7/92
118 0 8/4/92
118 0 8/3/92
118 5 8/14/93
118 5 8/4/94
118 0 8/20/92
118 15 8/27/92
118 95 8/26/92
118 15 8/25/92
118 0 8/24/92
118 0 8/18/92
118 5 8/21/92
118 5 7/24/93
118 0 7/28/93
118 5 7/29/93
118 0 7/23/93
118 0 7/26/93
118 0 7/31/93
118 0 8/6/93
118 0 8/5/93
118 0 8/2/93
19 70 9/10/96

19 55 7/21/92
19 10 7/22/92
19 5 7/23/92
19 200 7/24/92
19 5 7/25/92
19 15 7/27/92
19 5 7/28/92
19 0 7/29/92
19 30 7/30/92
19 25 7/31/92
19 0 6/14/93
19 15 6/16/93
19 10 6/28/93
19 5 7/2/93
19 25 6/30/93
19 0 6/19/93
19 10 7/20/92
19 60 6/29/93
19 5 8/26/91
19 0 6/24/93
19 25 6/25/93
19 15 6/26/93
19 933 6/29/89
19 108 6/23/90
19 33 6/13/90
19 10 6/14/90
19 110 6/15/90
19 30 6/16/90
19 328 6/19/90
19 40 8/29/91
19 33 6/22/90
19 375 8/17/89
19 33 6/25/90
19 3 6/26/90
19 20 6/27/90

19 93 6/29/90
19 858 6/30/90
19 75 7/2/90
19 293 6/21/90
19 33 8/7/89
19 60 7/19/89
19 228 7/24/89
19 63 7/25/89
19 33 7/26/89
19 670 7/31/89
19 243 8/1/89
19 140 6/12/90
19 1073 8/3/89
19 88 8/25/89
19 205 8/8/89
19 13 8/9/89
19 6750 8/14/89
19 4975 8/15/89
19 31400 8/16/89
19 213 10/1/90
19 130 8/2/89
19 15 7/8/92
19 120 7/3/90
19 180 8/23/91
19 240 8/24/91
19 25 6/21/93
19 80 8/27/91
19 40 8/30/91
19 2050 8/21/91
19 55 7/7/92
19 1550 8/20/91
19 40 7/9/92
19 155 7/10/92
19 80 7/13/92

19 265 7/14/92
19 395 7/15/92
19 105 7/17/92
19 335 7/6/92
19 870 10/18/90
19 80 7/18/92
19 30 10/2/90
19 5 10/4/90
19 100 10/9/90
19 12700 10/14/90
19 3550 10/15/90
19 1350 8/22/91
19 290 10/17/90
19 115 7/5/90
19 10 8/12/91
19 10 8/13/91
19 75 8/14/91
19 55 8/15/91
19 25 8/16/91
19 65 8/17/91
19 955 10/16/90
19 30 7/21/96
19 10 6/22/94
19 35 6/12/95
19 5 7/9/96
19 10 7/11/96
19 35 7/12/96
19 1670 7/14/96
19 530 7/15/96
19 775 7/15/96
19 1330 7/16/96
19 610 7/17/96
19 5 6/18/93
19 40 7/19/96
19 90 7/9/95

19 25 7/22/96
19 90 7/9/94
19 0 7/7/94
19 15 7/2/94
19 0 7/1/94
19 65 6/30/94
19 20 6/29/94
19 30 6/28/94
19 135 6/25/94
19 40 6/24/94
19 40 6/23/94
19 190 7/18/96
19 10 6/26/95
19 5 7/8/95
19 10 7/7/95
19 50 7/6/95
19 180 7/5/95
19 55 7/2/95
19 5 7/1/95
19 20 6/30/95
19 20 6/29/95
19 20 6/29/95
19 5 6/28/95
19 30 7/8/96
19 5 6/27/95
19 0 7/5/94
19 20 6/25/95
19 5 6/24/95
19 135 6/23/95
19 880 6/22/95
19 5 6/21/95
19 30 6/20/95
19 5 6/16/95
19 30 6/15/95

19 70 6/15/95
19 20 6/14/95
19 80 6/13/95
19 5 6/28/95
19 5 8/19/96
19 9800 7/25/96
19 45 6/21/94
19 5 7/8/94
19 225 7/27/96
19 310 7/29/96
19 175 7/30/96
19 585 8/2/96
19 875 7/24/96
19 245 8/1/96
19 370 7/26/96
19 25 8/27/96
19 10 8/28/96
19 10 8/29/96
19 105 8/30/96
19 10 9/3/96
19 55 9/4/96
19 80 9/9/96
19 4450 8/3/96
19 180 7/27/96
19 35 7/23/96
20 13 8/7/89
20 70 7/24/89
20 208 7/25/89
20 20 7/26/89
20 633 7/31/89
20 30 8/1/89
20 153 8/3/89
20 63 8/8/89
20 2275 8/15/89
20 118 8/17/89

20 25 8/2/89
20 17175 8/16/89
20 7525 8/14/89
20 3 8/9/89
81 30 8/15/92
81 0 8/14/93
81 10 8/21/92
81 5 8/20/92
81 100 8/19/92
81 15 8/18/92
81 8 8/13/90
81 3 8/7/90
81 0 8/11/93
81 5 8/14/90
81 21350 8/14/92
81 25 8/13/92
81 280 8/12/92
81 5 8/10/92
81 35 9/24/91
81 0 8/17/92
81 20 9/26/91
81 0 7/31/93
81 10 8/2/93
81 10 8/5/93
81 5 8/6/93
81 5 8/9/93
81 50 8/9/90
81 5 8/13/93
81 220 7/28/93
81 5 8/12/93
81 83 8/10/90
81 123000 7/23/93
81 5 8/8/90
81 30 7/24/93

81 105 7/25/93
81 0 7/29/93
81 15 8/3/90
81 130 8/20/94
81 5 8/27/92
81 10 8/2/94
81 0 8/3/94
81 0 8/4/94
81 0 8/5/94
81 20 8/6/94
81 105800 8/9/94
81 0 8/11/94
81 5 8/12/94
81 60 8/13/94
81 10 8/16/94
81 0 8/6/90
81 15 8/19/94
81 5 9/27/91
81 0 8/2/90
81 40 9/30/91
81 0 10/1/91
81 10 10/2/91
81 210 10/3/91
81 35 8/18/94
81 8 8/4/90
81 5 8/24/92
81 5 8/8/92
81 0 8/7/92
81 215 8/4/92
81 0 7/26/93
81 5 8/3/92
81 5 8/25/92
81 10 8/26/92
81 65 10/4/91

82 0 8/3/90
82 85 10/4/91
82 5 8/2/90
82 0 8/13/93
82 10 10/2/91
82 0 8/13/92
82 20 8/14/92
82 10 8/15/92
82 5 8/17/92
82 0 8/18/92
82 70 8/19/92
82 295 8/20/92
82 5 8/21/92
82 0 8/24/92
82 10 8/25/92
82 20 8/26/92
82 0 8/10/92
82 0 8/14/93
82 0 8/8/92
82 5 8/12/93
82 0 8/11/93
82 0 8/9/93
82 5 8/6/93
82 0 8/5/93
82 5 8/2/93
82 0 7/31/
93 82 5 7/29/
93 82 170 7/28/
93 82 5 7/26/93
82 115 7/25/93
82 30 7/24/93
82 640000 7/23/93
82 5 8/27/92

82 125 8/19/94
82 3 8/6/90
82 0 8/7/90
82 5 8/8/90
82 58 8/9/90
82 40 8/10/90
82 0 8/13/90
82 10 8/14/90
82 0 9/24/91
82 10 9/26/91
82 30 9/27/91
82 80 9/30/91
82 10 10/1/91
82 220 8/12/92
82 35 8/20/94
82 8 8/4/90
82 550 8/18/94
82 25 8/16/94
82 15 8/12/94
82 0 8/11/94
82 83600 8/9/94
82 25 8/6/94
82 10 8/5/94
82 0 8/4/94
82 10 8/3/94
82 0 8/2/94
82 10 8/3/92
82 195 8/4/92
82 0 8/7/92
82 290 10/3/91

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