Mathematics, 10.03.2020 08:08 ajm1132005

Men have head breadths that are normally distributed with a mean of 6.0 inches and a standard deviation of 1.0 inch. Engineers must consider the breadths of male heads when designing motorcycle helmets. Due to financial constraints, the helmets will be designed to fit all men except those with head breadths that are in the smallest 2.5% Find the minimum head breadth that the helmets will fit. Question 2 options:3

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Answer from: mem8163

For this answer, we omit the last part of the question "Question 2 options:3".

The minimum head breadth that the helmets will fit is 4.04 inches

Step-by-step explanation:

To answer the question, we need to find the value for which 2.5% of the cases are below it. To find it, we also need to consult the cumulative standard normal table and use the formula for z-score.

It is important to remember that we can find all probabilities for any normally distributed data using the standard normal table because we can 'transform' any raw score into a z-score, the value that this table uses to find probabilities. A z-score tells us how far is the raw score (and also the z-score) from the population mean.

The formula for z-scores is as follows:

$\\ z = \frac{x - \mu}{\sigma}$ (1)

Where

$\\ x\;is\;the\;raw\;score$ .

$\\ \mu\;is\;the\;population\;mean$ .

$\\ \sigma\;is\;the\;population\;standard\;deviation$ .

The minimum head breadth that the helmets will fit

To find the minimum head breadth, we can proceed as follows:

The smallest 2.5%

Since the normal distribution is symmetrical, the values for the smallest 2.5% and the biggest 2.5% in the distribution are at the same distance from the mean (although in the opposite direction), which is the key to solve this question.

However, we have a restriction here, namely, the cumulative standard normal table only gives positive values for the z-scores. We know that for values below the mean in the standard normal distribution, the z-scores are negative, and for those above the mean, they are positive. So, we need to find a value below the mean, that is, a negative value for the z-score, since we need to find the smallest 2.5%.

Then, as above mentioned, the values for the smallest 2.5% are at the same distance from the mean (but in the opposite direction) than the biggest 2.5%. Let us find the latter.

The biggest 2.5% are values where 97.5% (97.5 + 2.5 = 100) are below it. In the cumulative standard normal distribution, for a cumulative probability of 97.5% or 0.975, the associated value for z is 1.96. Thus, for the smallest 2.5%, the z-score is -1.96.

Using the formula (1), we can, therefore, find the raw score x for which 2.5% of the cases in the distribution are below it.

Then

$\\ \mu = 6.0\;inches$

$\\ \sigma = 1.0\;inch$

$\\ z = -1.96$

$\\ -1.96 = \frac{x - 6.0}{1.0}$

$\\ -1.96 * 1.0 = x - 6.0$

$\\ (-1.96 * 1.0) + 6.0 = x$

$\\ x = (-1.96 * 1.0) + 6.0$

$\\ x = 4.04\;inches$

Thus, the minimum head breadth that the helmets will fit is 4.04 inches. In other words, below this minimum head breadth no helmet will be produced because "the helmets will be designed to fit all men except those with head breadths that are in the smallest 2.5%".

We can see the values below x = 4.04 in the next graph.