Mathematics, 20.06.2020 23:57 anayamllr11

Engineers must consider the breadths of male heads when designing helmets. The company researchers have determined that the population of potential clientele have head breadths that are Normally distributed with a mean of 6.5 inches and a standard deviation of 1.2 inches. According to the 68-95-99.7 rule, we expect 95% of head breadths to be:.

Answers: 1

Show answers

Answers

Answer from: rowdycar313p0ao5k

"According to the 68-95-99.7 rule, we expect 95% [95.45%] of head breadths to be" between 4.1 inches and 8.9 inches.

Step-by-step explanation:

According to the 68-95-99.7 rule, approximately:

68% (more precisely, 68.27%) of the data from the normal distribution lie one standard deviation, $\\ \sigma$ , above and below the population mean, $\\ \mu$ .95% (more precisely, 95.45%) of the data lie two standard deviations, $\\ 2\sigma$ , above and below the population mean, $\\ \mu$ , and finally,99.7 (or more precisely, 99.73%) of the data lie three standard deviations, $\\ 3\sigma$ , above and below the population mean, $\\ \mu$ .

Then, if we have--from the question--that:

The random variable is head breadths.This variable follows a normal distribution.The population's mean for this distribution is $\\ \mu = 6.5$ inches.The population's standard deviation is $\\ \sigma =1.2$ inches.

We have to remember that two parameters characterize a normal distribution: the population's mean and the population's standard deviation. So, mathematically, the distribution we have from question is $\\ N(6.5, 1.2)$ .

For 95% (95.45%) of the head breadths, we expect that they are two standard deviations below and above the population's mean.

For solving this, we need to use the cumulative standard normal distribution (in case we need to find probabilities) and also use standardized values or z-scores:

$\\ z = \frac{x - \mu}{\sigma}$ [1]

A z-score "tells us" the distance from the mean in standard deviations units. If the z-score is positive, it is above the mean. If it is negative, it is below the mean.

Since 95% (95.45%) of the head breadths are two standard deviations (above and below the mean), we have (using [1]):

$\\ \pm2 = \frac{x - \mu}{\sigma}$

But we already know that $\\ \mu=6.5$ inches and $\\ \sigma=1.2$ inches.

Thus (without using units) for values above the population's mean:

$\\ 2 = \frac{x - 6.5}{1.2}$

Solving the equation for x, we multiply by 1.2 at each side of [1] :

$\\ 2 * 1.2 = \frac{x - 6.5}{1.2} * 1.2$

$\\ 2 * 1.2 = (x - 6.5)\frac{1.2}{1.2}$

$\\ 2 * 1.2 = (x - 6.5)*1$

$\\ 2 * 1.2 = x - 6.5$

Adding 6.5 at each side of the previous equation:

$\\ (2 * 1.2) + 6.5 = x - 6.5 + 6.5$

$\\ (2 * 1.2) + 6.5 = x + 0$

$\\ (2 * 1.2) + 6.5 = x$

Therefore, the raw value, x, in the distribution that is two standard deviations above the population's mean is:

$\\ x = (2 * 1.2) + 6.5$

$\\ x = 2.4 + 6.5$

$\\ x = 8.9$ inches.

For two standard deviations below the mean, we proceed in the same way:

$\\ -2 = \frac{x - 6.5}{1.2}$

$\\ -2*1.2 = x - 6.5$

$\\ (-2*1.2) + 6.5 = x$

$\\ x = (-2*1.2) + 6.5$

$\\ x = -2.4 + 6.5$

$\\ x = 4.1$ inches

Therefore, "according to the 68-95-99.7 rule, we expect 95% [95.45%] of head breadths to be" between 4.1 inches and 8.9 inches.

The graph below shows these values, and the shaded area represents 95% of the data, or, to be more precise, 95.45% (0.954499).

Engineers must consider the breadths of male heads when designing helmets. The company researchers h

Answer from: Quest

y + 11 = (1/4)(x + 16)

step-by-step explanation:

Answer from: Quest

answer: what is the conflict?

step-by-step explanation:

*what has happened to new york city?

*what is in the time capsule?

* what did the kidz see that gave them hope?

*what will they do with what they find?

* how will they save the city?

* what challenges will they face along the way?

*what big choice will they make at the climax of the story?

* what lesson do they learn at the end of the story?

Another question on Mathematics

Mathematics, 21.06.2019 17:30

When a rectangle is dilated, how do the perimeter and area of the rectangle change?

Answers: 2

Answer

Mathematics, 21.06.2019 20:10

A. use the formula for continuous compounding with the original example: $1000 invested at 2% for 1 year. record the amount to 5 decimal places. use a calculator. b. compare it to the result using the original compound interest formula with n = 365 calculated to 5 decimal places. which has a larger value? explain.

Answers: 1

Answer

Mathematics, 21.06.2019 20:30

The distance of a chord from the centre of a circle is 12cm and the lenght of the chord is 10cm. find the radius of the circle

Answers: 1

Answer

Mathematics, 21.06.2019 21:30

Are the corresponding angles congruent? explain why or why not.

Answers: 2

Answer

You know the right answer?

Engineers must consider the breadths of male heads when designing helmets. The company researchers h...