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Andrej [43]
3 years ago
15

Assume that weights of men are normally distributed with a mean of 170 lb and a standard deviation of 30 lb (approximate values

based on data from the National Health and Nutrition Examination Survey). What percentage of individual men have weights less than 185 lb? If samples of 36 men are randomly selected and the mean weight is computed for each sample, what percentage of the sample means are less than 185 lb?
a. 0.3156
b. 0.2611
c. 0.2312
d. 0.4602
Mathematics
1 answer:
Lubov Fominskaja [6]3 years ago
4 0

Answer:

a) P(X

b) P(\bar X

Step-by-step explanation:

1) Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

2) Part a

Let X the random variable that represent the weights of men of a population, and for this case we know the distribution for X is given by:

X \sim N(170,30)  

Where \mu=170 and \sigma=30

We are interested on this probability

P(X

And the best way to solve this problem is using the normal standard distribution and the z score given by:

z=\frac{x-\mu}{\sigma}

If we apply this formula to our probability we got this:

P(X

And we can find this probability on this way:

P(Z

3) Part b

The central limit theorem states that "if we have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed. This will hold true regardless of whether the source population is normal or skewed, provided the sample size is sufficiently large".

From the central limit theorem we know that the distribution for the sample mean \bar X is given by:

\bar X \sim N(\mu, \frac{\sigma}{\sqrt{n}})

P(\bar X

And using a calculator, excel or the normal standard table we have that:

P(Z

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Convert 2 inches into centimeters. Round your answer to the nearest tenth.
kobusy [5.1K]

Answer:

5.1 centimeters (5.08 not rounded)

Step-by-step explanation:

1 inch=2.54 centimeters

2.54*2=5.08

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3 years ago
How many 3 3/4 inch wires can be cut from a spool if wire that is 100 inches how much will be left over
Akimi4 [234]

How many 3\frac{3}{4} wires can be cut from a spool of wire that is 100 inches long?

How many inches will be left over?

Number of pieces of wires = length of spool of wire ÷ length of each piece of wire.

Number of pieces = 100 ÷ 3\frac{3}{4} = 100 ÷ \frac{15}{4}

That gives \frac{100 * 4}{15}

Simplifying gives \frac{400}{15} = \frac{80}{3}

This is equivalent to 26\frac{2}{3} pieces

So 26 pieces of wires can be cut from a spool and \frac{2}{3} of a piece will be left over.

6 0
3 years ago
Ashley has $9.05 in dimes and nickels. If she has a total of 108 coins, how many of each type does she have?
klemol [59]

Answer:

73\:\mathrm{dimes},\\35\:\mathrm{nickels}

Step-by-step explanation:

We can write the following system of equations:

\begin{cases}10d+5n=905,\\d+n=108,\\\end{cases}, where d is the number of dimes she has and n is the number of nickels she has.

Multiply the second equation by -5 and add both equations to solve for d:

\begin{cases}10d+5n=905,\\-5d-5n=-540,\\\end{cases}\\5d=365,\\d=\fbox{$73$}.

Plug d=73 into any equation to solve for n:

73+n=108, \\n=\fbox{$35$}

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2 years ago
Four groups of students are doing a probability experiment with a standard number cube to see how many times they roll a 4 out o
snow_lady [41]

Answer:

Group a, I did it

Step-by-step explanation:

The solution is Group A. As the number of trials get larger, experimental probability approaches theoretical probability. With a probability of 0.175 group A rolled a 4 14 out of 80 times.

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5 0
3 years ago
A piece of wire 30 m long is cut into two pieces. One piece is bent into a square and the other is bent into a circle.
antiseptic1488 [7]

Answer:

a) 0 m

b) 16.8 m

Step-by-step explanation:

A piece of wire, 30 m long, is cut in two sections: a and b. Then, the relation between a and b is:

a+b=30\\\\b=30-a

The section "a" is used to make a square and the section "b" is used to make a circle.

The section "a" will be the perimeter of the square, so the square side will be:

l=a/4

Then, the area of the square is:

A_s=l^2=(a/4)^2=a^2/16

The section "b" will be the perimeter of the circle. Then, the radius of the circle will be:

2\pi r=b=30-a\\\\r=\dfrac{30-a}{2\pi}

The area of the circle will be:

A_c=\pi r^2=\pi\left(\dfrac{30-a}{2\pi}\right)^2=\pi\left(\dfrac{900-60a+a^2}{4\pi^2}\right)=\dfrac{900-60a+a^2}{4\pi}

The total area enclosed in this two figures is:

A=A_s+A_c=\dfrac{a^2}{16}+\dfrac{900-60a+a^2}{4\pi}=\left(\dfrac{1}{16}+\dfrac{1}{4\pi}\right)a^2-\dfrac{60a}{4\pi}+\dfrac{900}{4\pi}

To calculate the extreme values of the total area, we derive and equal to 0:

\left(\dfrac{1}{16}+\dfrac{1}{4\pi}\right)a^2-\dfrac{60a}{4\pi}+\dfrac{900}{4\pi}\\\\\\\dfrac{dA}{da}=\left(\dfrac{1}{16}+\dfrac{1}{4\pi}\right)(2a)-\dfrac{60}{4\pi}+0=0\\\\\\\left(\dfrac{1}{8}+\dfrac{1}{2\pi}\right)a=\dfrac{15}{\pi}\\\\\\\dfrac{\pi+4}{8\pi}\cdot a=\dfrac{15}{\pi}\\\\\\\dfrac{\pi+4}{8}\cdot a=15\\\\\\a=15\cdot \dfrac{8}{\pi+4}\approx 16.8

We obtain one value for the extreme value, that is a=16.8.

We can derive again and calculate the value of the second derivative at a=16.8 in order to know if the extreme value is a minimum (the second derivative has a positive value) or is a maximum (the second derivative has a negative value):

\dfrac{d^2A}{da^2}=\left(\dfrac{1}{16}+\dfrac{1}{4\pi}\right)(2)-0=\dfrac{1}{8}+\dfrac{1}{2\pi}>0

As the second derivative is positive at a=16.8, this value is a minimum.

In order to find the maximum area, we analyze the function. It is a parabola, which decreases until a=16.8, and then increases.

Then, the maximum value has to be at a=0 or a=30, that are the extremes of the range of valid solutions.

When a=0 (and therefore, b=30), all the wire is used for the circle, so the total area is a circle, which surface is:

A=\pi r^2=\pi\left( \dfrac{30}{2\pi}\right)^2=\dfrac{900}{4\pi}\approx71.62

When a=30, all the wire is used for the square, so the total area is:

A=a^2/16=30^2/16=900/16=56.25

The maximum value happens for a=0.

3 0
3 years ago
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