Thank you for visiting In a shipment of 300 connecting rods the mean tensile strength is found to be 45 kpsi with a standard deviation of 5 kpsi a. This page is designed to guide you through key points and clear explanations related to the topic at hand. We aim to make your learning experience smooth, insightful, and informative. Dive in and discover the answers you're looking for!
Answer :
Answer:
a) Between 39 and 40 rods can be expected to have a strength less than 39.4 kpsi.
b) 260 rods are expected to have a strength between 39.4 and 60 kpsi
Step-by-step explanation:
Problems of normally distributed samples are solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
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.
In this problem, we have that:
[tex]\mu = 45, \sigma = 5[/tex]
(a) Assuming a normal distribution, how many rods can be expected to have a strength less than 39.4 kpsi?
The percentage of rods with a stength less than 39.4 is the pvalue of Z when X = 39.4. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{39.4 - 45}{5}[/tex]
[tex]Z = -1.12[/tex]
[tex]Z = -1.12[/tex] has a pvalue of 0.1314
13.14% of rods have a strength less than 39.4 kpsi.
Out of 300
0.1314*300 = 39.42
Between 39 and 40 rods can be expected to have a strength less than 39.4 kpsi.
(b) How many are expected to have a strength between 39.4 and 60 kpsi?
The percentage of rods with a stength in this interval is the pvalue of Z when X = 60 subtracted by the pvalue of Z when X = 39.4. So
X = 60
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{60 - 45}{5}[/tex]
[tex]Z = 3[/tex]
[tex]Z = 3[/tex] has a pvalue of 0.9987
X = 39.4
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{39.4 - 45}{5}[/tex]
[tex]Z = -1.12[/tex]
[tex]Z = -1.12[/tex] has a pvalue of 0.1314
0.9987 - 0.1314 = 0.8673
86.73% of the rods are expected to have a strength between 39.4 and 60 kpsi
Out of 300
0.8673*300 = 260
260 rods are expected to have a strength between 39.4 and 60 kpsi
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Rewritten by : Jeany
