Question

In: Math

The distribution of the number of people in line at a grocery store has a mean...

The distribution of the number of people in line at a grocery store has a mean of 3 and a variance of 9. A sample of the numbers of people in line in 50 stores is taken.

(a) Calculate the probability that the sample mean is more than 4? Round values to four decimal places.

(b) Calculate the probability the sample mean is less than 2.5. Round answers to four decimal places.

(c) Calculate the probability that the the sample mean differs from the population mean by less than 0.5. Round answers to four decimal places.

Please help, and show step by step. Thank you.

Solutions

Expert Solution

Population mean = 3

Population variance = 9

Population standard deviation = 3

Sample size, n = 50

For sample

Sample mean = Population mean = 3

Sample standard deviation, sd = Population standard deviation / sqrt(n) = 3/sqrt(50) = 0.4243

(a) Calculate the probability that the sample mean is more than 4?

Let, x_bar be the sample mean

P[ x_bar > 4 ] = P[ ( x_bar - mean )sd > ( 4 - mean )sd ]

P[ x_bar > 4 ] = P[ ( x_bar - 3 )0.4243 > ( 4 - 3 )0.4243 ]

P[ x_bar > 4 ] = P[ Z > 2.36 ]

P[ x_bar > 4 ] = 1 - P[ Z < 2.36 ]

P[ x_bar > 4 ] = 1 - 0.9909

P[ x_bar > 4 ] = 0.0091

(b) Calculate the probability the sample mean is less than 2.5.

Let, x_bar be the sample mean

P[ x_bar < 2.5 ] = P[ ( x_bar - mean )sd < ( 2.5 - mean )sd ]

P[ x_bar < 2.5 ] = P[ ( x_bar - 3 )0.4243 < ( 2.5 - 3 )0.4243 ]

P[ x_bar < 2.5 ] = P[ Z < -1.18 ]

P[ x_bar < 2.5 ] = 0.1190

(c) Calculate the probability that the sample mean differs from the population mean by less than 0.5

Let, x_bar be the sample mean

P[ 3 - 0.5 < x_bar < 3 + 0.5 ] = P[ 2.5 < x_bar < 3.5 ]

P[ 2.5 < x_bar < 3.5 ] = P[ ( 2.5 - mean )sd < ( x_bar - mean )sd < ( 3.5 - mean )sd ]

P[ 2.5 < x_bar < 3.5 ] = P[ ( 2.5 - 3 )0.4243 < ( x_bar - 3 )0.4243 < ( 3.5 - 3 )0.4243 ]

P[ 2.5 < x_bar < 3.5 ] = P[ -1.181 < Z < 1.18 ]

P[ 2.5 < x_bar < 3.5 ] = P[ Z < 1.18 ] - P[ Z < -1.18 ]

P[ 2.5 < x_bar < 3.5 ] = 0.8810 - 0.1190

P[ 2.5 < x_bar < 3.5 ] = 0.7620

milcah answered 3 months ago
Next > < Previous

Related Solutions

The distribution of the number of people in line at a grocery store checkout has a...
The distribution of the number of people in line at a grocery store checkout has a mean of 3 and a variance of 9. A sample of the numbers of people in 50 grocery store checkout lines is taken. (a) (3 points) What’s the probability that the sample mean is less than 4.5? Round your answer to four (4) decimal places. (b) (3 points) What’s the probability that the sample mean is more than 2.25? Round your answer to four...
A grocery store receipts show that Sunday customer purchases have a skewed distribution with a mean...
A grocery store receipts show that Sunday customer purchases have a skewed distribution with a mean of $35 and a standard deviation of $23. Suppose the store had 324 customers this Sunday A) Estimate the probability that the store's revenues were at least $12,200? B) If, on a typical Sunday, the store serves 324 customers, how much does the store take in on the worst 10% of such days?
Tom worked as a manager for a corner grocery store that employed five people. The grocery...
Tom worked as a manager for a corner grocery store that employed five people. The grocery store does not provide health insurance for its employees due to the expense. Tom, his wife, Mary and their three kids were uninsured. Tom’s wife did not want to incur any medical bills and ignored a mole on her chest. After many months of delay, Tom insisted that she see a dermatologist. She was diagnosed with malignant melanoma, which had metastasized. She died two...
A small grocery store has a single checkout line. On Saturdays, customers arrive at the checkout...
A small grocery store has a single checkout line. On Saturdays, customers arrive at the checkout on an average of one every 8 minutes. The cashier takes an average of 6 minutes to process a single customer. We assume that the service time is randomly distributed, and the customers arrive randomly. The store's owner believes that the amount of time that a customer has to wait hurts his business; he estimates that waiting time costs him $20 per customer-hour in...
Srini Nair owns and operates a grocery store in Fort McMurray. He has a number of...
Srini Nair owns and operates a grocery store in Fort McMurray. He has a number of employees including Ragini Sharma who is his assistant manager. Ragini has very limited authority to act on Srini’s behalf. One day while Srini was out of town, a distributor of produce offers Ragini a great deal on fiddleheads, a delicacy from New Brunswick. Due to the perishable nature of fiddleheads, the deal is only available for one day. As Srini’s grocery store caters to...
The upper boundaries for a distribution of waiting times (in seconds) in a grocery store aisle...
The upper boundaries for a distribution of waiting times (in seconds) in a grocery store aisle are 43, 54, 65, and 76. List the value for each lower boundary in this distribution. Lower boundary Upper boundary 76 65 54 43
Consider a Hotelling line of length 1. A grocery store is located at each endpoint. The...
Consider a Hotelling line of length 1. A grocery store is located at each endpoint. The grocery store at the west end point is owned and operated by Jack Donaghy and offers customers the use of a personal “grocery concierge”, free of charge, who assists the customer in shopping. The grocery store at the east endpoint is owned and operated by Milton Greene and does not offer its customers a grocery concierge. The utility of a consumer at location x...
he upper boundaries for a distribution of waiting times (in seconds) in a grocery store aisle...
he upper boundaries for a distribution of waiting times (in seconds) in a grocery store aisle are 44, 55, 66, and 77. List the value for each lower boundary in this distribution. Lower boundary Upper boundary upper boundaries are 77, 66, 55, 44
he upper boundaries for a distribution of waiting times (in seconds) in a grocery store aisle...
he upper boundaries for a distribution of waiting times (in seconds) in a grocery store aisle are 49, 60, 71, and 82. List the value for each lower boundary in this distribution.
The mean cost of the ingredients of a chicken permesan meal bought at the grocery store...
The mean cost of the ingredients of a chicken permesan meal bought at the grocery store is $15. Prices from a sample of 50 restaurants for a comparable meal is shown in spreadsheet PRICES. At an alpha level of .05, can we say that eating chicken parmensan at a restaurant is cheaper? Formulate your hypotheses and solve the problem. 12.7 12.31 13.51 13.97 9.24 11.4 13.73 13.47 11.62 11.69 14.32 12.42 18.49 11.66 16.59 14.62 14.77 14.06 14.66 9.22 12.64...
ADVERTISEMENT
Subjects
ADVERTISEMENT
Latest Questions
ADVERTISEMENT