Comparing observed data to a theoretical model. (a) Consider the simplistic mode

Question

Comparing observed data to a theoretical model. (a) Consider the simplistic model that human births are evenly distributed over the 12 calendar months (each person has an equal chance of being born in each month). If a person is randomly selected, say from the phone directory, what is the probability that his or her birthday would be i. In a winter month (start of November-end of February)? ii, Not over the summer where summer is start of May-end of August? (b) The following record shows a classification of births (in thousands) in the United States. Calculate the relative frequency of births for each month and comment on the plausibility of a uniform probability model (each month has the same probability) Jan. Feb. March ApMay June | July Aug. |Sept. | Oct. | Nov. | Dec. |Total 331.5309.5 6 349.3 332.5 346.3 350.9 357.1 369.3 363.4344.6 335.7 348.3 4,138

Explanation / Answer

(a)

(i) We have the general probability formula as :

Probability = Number of favourable outcomes / Total number of outcomes

Now , Favourable outcomes in our case will be birth in month of november , december , january and february.

So, Number of favourable outcomes = 4

and since there are 12 months in a year

So, Total number of outcomes = 12

Using the formula ,

Probability that his or her birthday would be in a winter month = Number of favourable outcomes / Total number of outcomes

= 4 /12 = 1 / 3 = 0.333

Probability that his or her birthday would be in a winter month = 0.333

(ii) Now , Favourable outcomes in our case will be birth in month of september, october , november , december , january , february, march and april

So, Number of favourable outcomes = 8

and since there are 12 months in a year

So, Total number of outcomes = 12

Using the formula ,

Probability that his or her birthday would be not over summer = Number of favourable outcomes / Total number of outcomes

= 8 /12 = 2 / 3 = 0.667

Probability that his or her birthday would be not over summer = 0.667

(b)

Relative frequency of births for each month can be calculated by dividing each month's frequency by total sum of frequencies.

For example for Jan month , Relative frequency = 331.5 / 4138.5 = 0.08

For feb month , Relative frequency = 309.6 / 4138.5 = 0.07

Annd similarly we can calculate for other months also.

So, we obtain the Relative frequency for each month as :

Now , we can see from the above table that relative frequency is almost same for each month which indicates plausibility of a uniform probability model.

Jan Feb March April May June July Aug. Sept. Oct. Nov. Dec. Total Frequency 331.5 309.6 349.3 332.5 346.3 350.9 357.1 369.3 363.4 344.6 335.7 348.3 4138.5 Relative frequency 0.08 0.07 0.08 0.08 0.08 0.08 0.09 0.09 0.09 0.08 0.08 0.08 1

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