Evaluate the expected costs of a refund program using probability calculations.

Firestone will refund customers if the snowfall in their area is less than average over the coming year according to a scheme.

The goal is to work outthe probability of having to award refunds in this area. From these probabilities, you can also calculate the expected cost of the refund per dollar in this area

Firestone uses the average annual snowfall in the last 10 years to award refunds. For the Toronto area, the 10-year average in the advertisement is 130.9 cm

1. First, you need to estimate the distribution of annual snowfall in Toronto using the historical data from 1940 to 1982 (the last column named "total" in the data set).
2. plot the histogram
3. Second, use the sample of annual snowfall to estimate the parameters of the Normal distribution.
4. Calculate the sample average and use it as an estimate for the actual mean of annual snowfall.
5. Also, calculate the sample standard deviation and use is as an estimate for the actual standard deviation of annual snowfall.
6. What are the estimates for the mean and standard deviation of annual snowfall in Toronto based on the sample of annual snowfall in the area from 1940 to 1982?
7. Calculate the expected cost of refund per dollar in Toronto in the coming year using the estimates for the mean and standard deviation of annual snowfall.
8. How would the refund probabilities be affected if the true standard deviation of annual snowfall were higher than the sample standard deviation? What if the true standard deviation were lower?
9. Evaluate (qualitatively) the value of the refund program to the consumer buying a snow tire in Toronto and to Firestone. Who is the major winner?

Year	Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec	Total
1940	29.7	38.9	35.1	10.4	0	0	0	0	0	0	42.2	11.9	168.2
1941	41.7	27.4	26.2	0	0	0	0	0	0	0	0	16	111.3
1942	11.9	33.5	14.2	14	0	0	0	0	0	0	14.5	37.1	125.2
1943	68.3	20.3	23.9	20.3	0	0	0	0	0	0	4.1	2.8	139.7
1944	6.6	49.5	30	17	0	0	0	0	0	0	10.7	92.5	206.3
1945	45.7	30.2	3.3	10.9	0	0	0	0	0	0	12.7	18.5	121.3
1946	56.6	54.6	1.5	3.3	0	0	0	0	0	0	13	24.1	153.1
1947	59.9	25.1	36.6	12.7	0.5	0	0	0	0	0	13	34.5	182.3
1948	37.1	48.5	16.3	0.3	0	0	0	0	0	0	0	42.2	144.4
1949	21.8	30.5	32.5	1.3	0	0	0	0	0	0	22.6	10.9	119.6
1950	41.7	70.9	29.7	1.8	0	0	0	0	0	0	57.2	3.6	204.9
1951	28.7	20.8	23.9	0	0	0	0	0	0	0	34	71.1	178.5
1952	45.5	19.1	7.9	0.3	0	0	0	0	0	2.3	0	9.1	84.2
1953	16	19.1	0.3	0	0	0	0	0	0	0	16.8	14.5	66.7
1954	37.1	36.8	30.5	0	0	0	0	0	0	0	2.8	24.1	131.3
1955	31.2	35.8	32	0	0	0	0	0	0	0	6.1	44.7	149.8
1956	36.3	30	44.5	12.2	0	0	0	0	0	0	7.6	22.1	152.7
1957	42.2	31	11.9	17.8	0	0	0	0	0	0	0.5	11.9	115.3
1958	32.8	19.6	10.2	4.6	0	0	0	0	0	0	28.2	25.4	120.8
1959	28.2	46.7	36.8	0	0.5	0	0	0	0	0	15.5	21.3	149
1960	63.5	63	42.4	5.3	0	0	0	0	0	0	1.8	14.7	190.7
1961	26.4	30.2	42.2	15	0	0	0	0	0	0	3.8	15.2	132.8
1962	26.7	62	2.5	5.3	0	0	0	0	0	2	1.8	24.4	124.7
1963	19.6	16.3	20.1	13.2	1.8	0	0	0	0	0	1.3	49.5	121.8
1964	15.7	33.8	32.3	10.2	0	0	0	0	0	0	3.3	23.9	119.2
1965	55.4	41.9	47	10.4	0	0	0	0	0	1.3	3.6	15	174.6
1966	70.9	13.5	13.5	24.6	1.3	0	0	0	0	0	9.4	20.3	153.5
1967	49	48.3	28.7	16.3	0.5	0	0	0	0	0	2.3	16	161.1
1968	51.1	7.6	39.9	0	0	0	0	0	0	0	7.6	50.5	156.7
1969	15	14	9.9	0	0	0	0	0	0	12.2	3.8	34.3	89.2
1970	29.5	15.5	16.3	2.8	0	0	0	0	0	0	4.6	67.8	136.5
1971	33.5	42.2	38.4	1.3	0	0	0	0	0	0	14.7	33.8	163.9
1972	31	47.5	40.9	11.9	0	0	0	0	0	0	14.2	49.5	195
1973	10.4	24.9	14.2	1.3	0	0	0	0	0	0	3	58.4	112.2
1974	30.5	25.7	16.5	0.8	0	0	0	0	0	0	1.5	27.4	102.4
1975	13.2	29	26.4	24.4	0	0	0	0	0	0	0.8	64	157.8
1976	53.1	4.8	37.8	10.2	0	0	0	0	0	0	3	40.9	149.8
1977	70.8	6.2	22.6	1.9	0	0	0	0	0	0	5.8	67.6	174.9
1978	63.4	28.3	11.4	0.8	0	0	0	0	0	0	13.4	14.2	131.5
1979	55.4	26.2	1.2	37.6	0	0	0	0	0	0	0.8	56.3	177.5
1980	10.8	14.9	38.1	0.2	0	0	0	0	0	0	10.9	28.3	103.2
1981	37.1	33.3	23.6	7.4	0.3	0	0	0	0	0.5	10.2	28.7	141.1
1982	36.4	28.5	24.7	7.6	0.1	0	0	0	0	0.6	7.4	33.9	139.2
1983	10.8	9.6	18.6	1.6	0