Final answer:
a.- The probability of x = 0 is 264/11014 = 0.023.
-The probability of x = 1 is 365/11014 = 0.033.
-The probability of x = 2 is 193/11014 = 0.018.
-The probability of x = 3 is 91/11014 = 0.008.
-The probability of x = 4 or more is 101/11014 = 0.009.
b. The expected value of x is 0.155
c. The variance of x is 0.460.
d. The expected value of y is 0.170
Step-by-step explanation:
(a) To develop a probability distribution for x, we need to calculate the probabilities for each value of x based on the number of responses.
- - For x = 0, the number of responses is 264. Therefore, the probability of x = 0 is 264/11014 = 0.023.
- - For x = 1, the number of responses is 365. So, the probability of x = 1 is 365/11014 = 0.033.
- - For x = 2, the number of responses is 193. Thus, the probability of x = 2 is 193/11014 = 0.018.
- - For x = 3, the number of responses is 91. Hence, the probability of x = 3 is 91/11014 = 0.008.
- - For x = 4 or more, the number of responses is 101. Consequently, the probability of x = 4 or more is 101/11014 = 0.009.
(b) To compute the expected value of x, we need to multiply each value of x by its corresponding probability and then sum them up.
E(x) = 0 * 0.023 + 1 * 0.033 + 2 * 0.018 + 3 * 0.008 + 4 * 0.009 = 0.155.
(c) To calculate the variance of x, we need to find the squared deviation of each value of x from the expected value, multiply it by its corresponding probability, and then sum them up.
var(x) = (0 - 0.155)² * 0.023 + (1 - 0.155)² * 0.033 + (2 - 0.155)² * 0.018 + (3 - 0.155)² * 0.008 + (4 - 0.155)²* 0.009 = 0.460.
(d) To compute the expected value of y, which represents the number of cups of coffee consumed on an average day for adults who drink at least one cup, we need to exclude the probability of x = 0 from the calculations.
E(y) = (1 * 0.033 + 2 * 0.018 + 3 * 0.008 + 4 * 0.009) / (1 - 0.023) = 0.170.
Your question is incomplete, but most probably the full question was:
In an annual poll about consumption habits, telephone interviews were concluded for a random sample of 1,1014 adults aged 18 and over. One of the questions was, ``How many cups of coffee, if any, do you drink on an average day?
`The following table shows the results obtained.
Number of cups per day Number of responses
0 264
1 365
2 193
3 91
4 or more 101
Define a random variable x= number of cups of coffee consumed on an average day. Let x=4 represent four or more cups.
(a) Develop a probability distribution for x. (Round your answers to three decimal places.)
(b) Compute the expected value of x. (Round your answer to three decimal places.)
E(x)=
(c) Compute the variance of x. (Round your answer to three decimal places.)
var(x)=
(d) Suppose we are only interested in adults who drink at least one cup of coffee on an average day. For this s e number of cups of coffee consumed on an average day. Compute the expected value of y. (Round your answer to three decimal places.)
E(y)=