Let x be a random variable that represents checkout time (time spent in the actual checkout process) in minutes in the express lane of a large grocery. Based on a consumer survey, the mean of the x distribution is about ? = 2.7 minutes, with standard deviation ? = 0.6 minute. Assume that the express lane always has customers waiting to be checked out and that the distribution of x values is more or less symmetric and mound-shaped. What is the probability that the total checkout time for the next 30 customers is less than 90 minutes? Let us solve this problem in steps.
Let x be a random variable that represents checkout time (time spent in the actual checkout process) in minutes in the express lane of a large grocery. Based on a consumer survey, the mean of the x distribution is about ? = 2.7 minutes, with standard deviation ? = 0.6 minute. Assume that the express lane always has customers waiting to be checked out and that the distribution of x values is more or less symmetric and mound-shaped. What is the probability that the total checkout time for the next 30 customers is less than 90 minutes? Let us solve this problem in steps.
Let x be a random variable that represents checkout time (time spent in the actual checkout process) in minutes in the express lane of a large grocery. Based on a consumer survey, the mean of the x distribution is about ? = 2.7 minutes, with standard deviation ? = 0.6 minute. Assume that the express lane always has customers waiting to be checked out and that the distribution of x values is more or less symmetric and mound-shaped. What is the probability that the total checkout time for the next 30 customers is less than 90 minutes? Let us solve this problem in steps.
Let x be a random variable that represents checkout time (time spent in the actual checkout process) in minutes in the express lane of a large grocery. Based on a consumer survey, the mean of the x distribution is about ? = 2.7 minutes, with standard deviation ? = 0.6 minute. Assume that the express lane always has customers waiting to be checked out and that the distribution of x values is more or less symmetric and mound-shaped. What is the probability that the total checkout time for the next 30 customers is less than 90 minutes? Let us solve this problem in steps.
Transcribed Image Text:Let x be a random variable that represents checkout time (time spent in the actual checkout process) in minutes in the express lane of a large grocery. Based on a consumer survey, the
mean of the x distribution is about u = 2.7 minutes, with standard deviation G = 0.6 minute. Assume that the express lane always has customers waiting to be checked out and that the
distribution of x values is more or less symmetric and mound-shaped. What is the probability that the total checkout time for the next 30 customers is less than 90 minutes? Let us solve
this problem in steps.
n USE SALT
(a) Let x, (for i = 1, 2, 3, ..., 30) represent the checkout time for each customer. For example, x, is the checkout time for the first customer, x, is the checkout time for the
second customer, and so forth. Each x; has mean u = 2.7 minutes and standard deviation o = 0.6 minute. Let w = x, + x, +... + x20: Explain why the problem is asking us to
compute the probability that w is less than 90.
O w is the total waiting time for all customers, so we want to find P(w < 90).
O w is the average of the waiting times for 30 customers, so we want to find P(w < 90).
O w is the sum of the waiting times for 30 customers, so we want to find P(w < 90).
O w is the average waiting time for 30 customers, so we want to find P(w < 90).
(b) Use a little algebra and explain why w < 90 is mathematically equivalent to w/30 < 3.
O subtract 30
O subtract 90
O divide by 3
O add 30
O divide by 30
Since w is the total of the 30 x values, then w/30 = x. Therefore, the statement x < 3 is equivalent to the statement w < 90. From this we conclude that the probabilities P(x < 3)
and P(w < 90) are equal.
(c) What does the central limit theorem say about the probability distribution of x? Is it approximately normal? What are the mean and standard deviation of the x distribution?
O The probability distribution of x is approximately normal with mean u- = 2.7 and standard deviation o = 0.6.
O The probability distribution of x is not normal.
O The probability distribution of x is approximately normal with mean u, = 2.7 and standard deviation o, = 0.02.
O The probability distribution of x is approximately normal with mean u, = 2.7 and standard deviation o = 0.11.
(d) Use the result of part (c) to compute P(x < 3). (Round your answer to three decimal places.)
What does this result tell you about P(w < 90)? (Round your answer to three decimal places.)
The probability that the total checkout time for the next 30 customers is less than 90 minutes is
Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...
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