I have to build a box for my scouting project with a volume of 36 cubic centimeters. The base of the box is 6 cm long and 2 cm w
1 answer:
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Answer:
a) P(12.99 ≤ X ≤ 13.01) = 0.3840
b) P(X ≥ 13.01) = 0.3075
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the cental limit theorem.
Normal probability distribution
Problems of normally distributed samples are solved using the z-score formula.
In a set with mean and standard deviation
, the zscore of a measure X is given by:
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean and standard deviation
, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
and standard deviation
.
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
In this problem, we have that:
(a) Calculate P(12.99 ≤ X ≤ 13.01) when n = 16.
Here we have
This probability is the pvalue of Z when X = 13.01 subtracted by the pvalue of Z when X = 12.99.
X = 13.01
By the Central Limit Theorem
has a pvalue of 0.6915
X = 12.99
has a pvalue of 0.3075
0.6915 - 0.3075 = 0.3840
P(12.99 ≤ X ≤ 13.01) = 0.3840
(b) How likely is it that the sample mean diameter exceeds 13.01 when n = 25?
P(X ≥ 13.01) =
This is 1 subtracted by the pvalue of Z when X = 13.01. So
has a pvalue of 0.6915
1 - 0.6915 = 0.3075
P(X ≥ 13.01) = 0.3075
Given:
Product A requires 7 board feet of wood and 4 lbs of wicker.
Product B requires 5 board feet of wood and 5 lbs of wicker.
Product C requires 4 board feet of wood and 3 lbs of wicker.
Available wood = 3000 board feet
Available wicker = 1400 lbs
Profit margin of A = $35 per unit
Profit margin of B = $42 per unit
Profit margin of C = $20 per unit
To find:
The linear programming problem for given situation.
Solution:
Let the number of units produced of products A, B and C are x, y and z respectively.
Product A Product B Product C Total
Board feet of wood 7 5 4 3000
wicker 4 5 3 1400
Objective function: Maximize
s.t.,
Board feet of wood :
Wicker :
Number of units cannot be negative. So, .
Therefore, the required LPP is
Maximize
s.t.,