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3 years ago

6

a physician wants to estimate with 92% confidence the average height of 6 year old boys in his community he wants his margin of

error to be 0.5 in the population of these heights is normally distributed with a standard deviation of 1.8 inches how many 6 year old boy should he use in his sample

1 answer:

iris [78.8K]3 years ago

5 0

Answer:

n=40 boys

Step-by-step explanation:

-Margin of error is given by the formula:

$E=z\frac{\sigma}{\sqrt{n}}$

-The z score for a probability of 92% is 1.75

#We make n the subject of the formula and substitute at a 92% confidence:

$E=z\frac{\sigma}{\sqrt{n}}\\\\\sqrt{n}=\frac{z\sigma}{E}\\\\n=[\frac{z\sigma}{E}]^2\\\\\therefore n=[\frac{1.75\times 1.8}{0.5}]^2\\\\=39.69\approx 40$

Hence, he needs approximately 40 boys for the study

You might be interested in

The amount of coffee that a filling machine puts into an 8-ounce jar is normally distributed with a mean of 8.2 ounces and a sta

nordsb [41]

Answer:

73.30% probability that the sampling error made in estimating the mean amount of coffee for all 8-ounce jars by the mean of a random sample of 100 jars will be at most 0.02 ounce

Step-by-step explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal probability distribution

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$ , the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

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.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean $\mu$ and standard deviation $\sigma$ , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$ .

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

In this problem, we have that:

$\mu = 8.2, \sigma = 0.18, n = 100, s = \frac{0.18}{\sqrt{100}} = 0.018$

What is the probability that the sampling error made in estimating the mean amount of coffee for all 8-ounce jars by the mean of a random sample of 100 jars will be at most 0.02 ounce?

This is the pvalue of Z when X = 8.2 + 0.02 = 8.22 subtracted by the pvalue of Z when X = 8.2 - 0.02 = 8.18. So

X = 8.22

$Z = \frac{X - \mu}{\sigma}$

By the Central Limit Theorem

$Z = \frac{X - \mu}{s}$

$Z = \frac{8.22 - 8.2}{0.018}$

$Z = 1.11$

$Z = 1.11$ has a pvalue of 0.8665

X = 8.18

$Z = \frac{X - \mu}{s}$

$Z = \frac{8.18 - 8.2}{0.018}$

$Z = -1.11$

$Z = -1.11$ has a pvalue of 0.1335

0.8665 - 0.1335 = 0.7330

73.30% probability that the sampling error made in estimating the mean amount of coffee for all 8-ounce jars by the mean of a random sample of 100 jars will be at most 0.02 ounce

8 0

4 years ago

2. Find the volume of the<br> cylinder.<br> 3.5 ft.<br> r= 2 ft.

lyudmila [28]

Answer:

43.96

(i used 3.14 for pie)

Step-by-step explanation:

4 0

3 years ago

If the terminal point of e is (0, -1), what is tan e?<br> A. O<br> B. 1<br> C. Undefined<br> D. -1

Leni [432]

The correct answer is C undefined

7 0

3 years ago

The Johnsons spent a total of $69.96 for dinner at their favorite restaurant last night. If this cost included a 20% tip, how

enyata [817]

It would be 55.97

To find this you find what 20% of 69.96 is and subtract that number from 69.96 (in this case that number would be 13.99) So you would find your answer of 55.97

5 0

3 years ago

The total surface area of a closed cylinder is

Degger [83]

The maximum volume of the cylinder is 27147.355 at the maximum point $r = \frac{50}{\sqrt{3} }$ .

<h3>How do you find the maximum volume of the cylinder?</h3>

The formula for the volume of the cylinder v = $\pi$ $r^{2}$ h, To find the maximum volume of the cylinder we apply the condition of maxima $\frac{\mathrm{d} v}{\mathrm{d} r}$ = 0.

Let r cm be the radius and h cm be the height of the closed cylinder.

Then, Total surface area of the cylinder = $2\pi r(r+h)$

$5000$ = $2\pi r^{2} +2\pi rh$

$h$ = $\frac{5000-2\pi r^{2} }{2\pi rh}$ .............(1)

Volume of the cylinder v = $\pi r^{2} h$

Substitute the value of $h$ in the above equation

$\Rightarrow$ = $\pi r^{2}$ × $\left [ \frac{5000-2\pi r^{2}}{2\pi r} \right ]$

$\Rightarrow$ = $\frac{r}{2}$ × $(5000-2\pi r^{2} )$

$\Rightarrow$ v = $2500r-\pi r^{3}$ ..............(2)

Now, for the maximum volume of the cylinder $\frac{\mathrm{d} v}{\mathrm{d} r}$ = 0

$\Rightarrow$ $\frac{\mathrm{d} (2500r-\pi r^{3})}{\mathrm{d} r} = 0$

$\Rightarrow$ $3\pi r^{2} = 2500$

$\Rightarrow$ $r^{2} = \frac{2500}{3\pi }$

$\Rightarrow$ $r = \frac{50}{\sqrt{3\pi } }$

Volume is maximum for $r = \frac{50}{\sqrt{3\pi } }$

Then, v = $2500r-\pi r^{3}$

$\Rightarrow$ = $2500 \frac{50}{\sqrt{3\pi } } -\pi (\frac{50}{\sqrt{3\pi } } )^{3}$

$\Rightarrow$ = $\frac{125000}{\sqrt{3\pi } } - \frac{125000}{3\sqrt{3\pi } }$

$\Rightarrow$ = $\frac{125000}{\sqrt{3\pi } }$ × $\frac{2}{3}$

$\Rightarrow$ = $\frac{250000}{3\sqrt{3\pi } }$

$\Rightarrow$ v = 27147.355

Hence, The maximum volume of the cylinder is 27147.355 at the maximum point $r = \frac{50}{\sqrt{3} }$ .

To learn more about total surface area and volume of the cylinder from the given link

brainly.com/question/16095729

#SPJ4

3 0

2 years ago