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Mean, Standard Deviation & Probability Distribution

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Find the mean, standard deviation and probability distribution for the given problems.

1. The average profit per unit for Product A is $5.32, Product B is $7.98, and Product C is $9.69. If the percentages of each sold are 36, 29, and 35 respectively, what is the weighted mean profit per unit?

2. A fleet of cabs for a small cab company consists of five units. The number of cabs in service varies over time and is a random variable. Daily log records show the probability distribution for this random variable as given below:

# in Service Probability
x P(x)
0 .05
1 .15
2 .16
3 .25
4 .21
5 .18

Find the mean and standard deviation of number of units in service.

3. The Southway National Bank surveyed the status of student accounts and found that the average overdraft was $21.22 with a standard deviation of $5.49. If the distribution is normal, find the probability of a student being overdrawn by more than $18.75.

4. An efficiency expert makes periodic checks for weighting errors for a long distance shipping firm. The expert inspects for errors in weighing, in recording weights, and errors in processing the bills of lading. Based on past records the number of weekly errors for all shipments averages 5.3 with a standard deviation of 1.23 and the frequency histogram approximates a normal distribution. Suppose x is the number is the number of weighing errors that will occur next week. Compute the approximate probability for x=3.

5. An elementary school teacher learned that 40% of school age children have at least 3 cavities.
a) If the standard deviation is 2.684, how many students would she expect to find her class of 30 who have at least 3 cavities?

Determine the probability that more than 20 students in her class will have 3 cavities.

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The expert examines the mean, standard deviation and probability distribution. The solution answers the question(s) below.

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1. The average profit per unit for Product A is $5.32, Product B is $7.98, and Product C is $9.69. If the percentages of each sold are 36, 29, and 35 respectively, what is the weighted mean profit per unit?

Solution. The weighted mean profit per unit is

5.32*36%+7.98*29%+9.69*35%=1.9152+2.3142+3.3915=7.6209

2. A fleet of cabs for a small cab company consists of five units. The number of cabs in service varies over time and is a random variable. Daily log records show the probability distribution for this random variable as given below:

# in Service Probability
x P(x)
0 .05
1 .15
2 .16
3 .25
4 .21
5 .18

Find the mean and standard deviation of number of units in service.
Solution. Denote the mean and standard deviation of X by E(X) and D(X) respectively. Then, by a formula of E(X)= we have
E(X)=0*0.05+1*0.15+2*0.16+3*0.25+4*0.21+5*0.18=0.15+0.32+0.75+0.84+0.9=2.96
By a formula ...

Solution provided by:
Changping Wang, MA
Education
  • BSc , Wuhan Univ. China
  • MA, Shandong Univ.
Recent Feedback
  • "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
  • "excellent work"
  • "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
  • "Thank you"
  • "Thank you very much for your valuable time and assistance!"
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