A survey of 200 shoppers was conducted to investigate how often people buy wine at a certain supermarket. Each shopper was asked whether they had bought any wine in each of the last 4 weeks. Let x denote the number of weeks when a shopper bought wine and fdenote the number of shoppers who bought wine in x weeks. If a shopper had not bought any wine then x was set to zero. Results are given in the first two rows of the following table. Note that f= 34 shoppers did not buy any wine, f= 40 shoppers bought wine only in 1 of those 4 weeks and so on. 10 1 2 3 Total 34 40 64 56 200 Pi 0.0915 0.2995 0.3675 0.2005 0.0410 It has been suggested that these data are binomially distributed with n - 4 and some (unknown) parameter p, which is the probability that a randomly selected shopper will buy wine in any given week. Answer the following questions. a. Calculate the sample mean and the sample variance for these data. b. Estimate the parameter p for the suggested binomial distribution. c. Write down the formula to calculate the binomial probability P( X = 3) - 0.2005 (4dp), as given in the table. Show clearly how to use the answer from part (b) to calculate the probability given here. d. Use the p;values given in the table, which were obtained as in part (c), to calculate corresponding expected values, ej. Give your expected values to 1 decimal place.
A survey of 200 shoppers was conducted to investigate how often people buy wine at a certain supermarket. Each shopper was asked whether they had bought any wine in each of the last 4 weeks. Let x denote the number of weeks when a shopper bought wine and fdenote the number of shoppers who bought wine in x weeks. If a shopper had not bought any wine then x was set to zero. Results are given in the first two rows of the following table. Note that f= 34 shoppers did not buy any wine, f= 40 shoppers bought wine only in 1 of those 4 weeks and so on. 10 1 2 3 Total 34 40 64 56 200 Pi 0.0915 0.2995 0.3675 0.2005 0.0410 It has been suggested that these data are binomially distributed with n - 4 and some (unknown) parameter p, which is the probability that a randomly selected shopper will buy wine in any given week. Answer the following questions. a. Calculate the sample mean and the sample variance for these data. b. Estimate the parameter p for the suggested binomial distribution. c. Write down the formula to calculate the binomial probability P( X = 3) - 0.2005 (4dp), as given in the table. Show clearly how to use the answer from part (b) to calculate the probability given here. d. Use the p;values given in the table, which were obtained as in part (c), to calculate corresponding expected values, ej. Give your expected values to 1 decimal place.
A First Course in Probability (10th Edition)
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ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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Transcribed Image Text:A survey of 200 shoppers was conducted to investigate how often people buy wine at a certain supermarket.
Each shopper was asked whether they had bought any wine in each of the last 4 weeks.
Let x denote the number of weeks when a shopper bought wine and fdenote the number of shoppers who
bought wine in x weeks. If a shopper had not bought any wine then x was set to zero. Results are given in the first
two rows of the following table. Note that f = 34 shoppers did not buy any wine, f = 40 shoppers bought wine
only in 1 of those 4 weeks and so on.
2
3
4
Total
34
40
64
56
6
200
Pi
0.0915
0.2995
0.3675
0.2005
0.0410
It has been suggested that these data are binomially distributed with n = 4 and some (unknown) parameter
p, which is the probability that a randomly selected shopper will buy wine in any given week.
Answer the following questions.
a. Calculate the sample mean and the sample variance for these data.
b. Estimate the parameter p for the suggested binomial distribution.
c. Write down the formula to calculate the binomial probability P( X = 3) = 0.2005 (4dp), as given in
the table. Show clearly how to use the answer from part (b) to calculate the probability given here.
d. Use the p;values given in the table, which were obtained as in part (C) to calculate corresponding
expected values, ej. Give your expected values to 1 decimal place.
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