Steel rods are manufactured with a mean length of 25 centimeters (cm). Because of variability in the manufacturing process, the lengths of the rods are approximately normally distributed, with a standard deviation of 0.07 cm. Any rods that are shorter than 24.85 cm or longer than 25.15 cm are discarded. What proportion of rods will be discarded

3.24% of rods will be discarded

Step-by-step explanation:

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.

In this problem, we have that:

What proportion of rods will be discarded

Shorter than 24.85.

pvalue of Z when X = 24.85

has a pvalue of 0.0162

Longer than 25.15

1 subtracted by the pvalue of Z when X = 25.15

has a pvalue of 0.9838

1 - 0.9838 = 0.0162

2*0.0162 = 0.0324

3.24% of rods will be discarded

The proportion of rods will be discarded is 0.03236

Step-by-step explanation:

From the question;

mean; μ = 25 cm

Standard deviation; σ = 0.07 cm)

Now, this is a normal distribution problem where we are supposed to find the Z-score.

Z is given by;

Z = (X - μ)/0.07

Applying this to the question of, rods that are shorter than 24.85 cm or longer than 25.15 cm are discarded. We arrive at;

P(X<24.85) + P(X>25.15)

This will give us;

P(z < [(24.85 - 25)/0.07)]) + P(z > [(25.15 - 25)/0.07)])

= P(z < (-2.143)) + P(z > 2.14)

This now transforms to;

= P(z ≤ (-2.143)) + 1 - P(z ≤ 2.143)

From standard normal table which i have attached,

P(z ≤ (-2.143)) = 0.01618 and

P(z ≤ 2.143) = 0.98382

Thus,

P(X<24.85) + P(X>25.15) = 0.01618 + 1 - 0.98382 = 0.03236

a variable is any characteristics, number, or quantity that can be measured or counted. a variable may also be called a data item. hope i !

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