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densk [106]
3 years ago
13

A news report states that the 95% confidence interval for the mean number of daily calories consumed by participants in a medica

l study is (2080, 2260). Assume the population distribution for daily calories consumed is normally distributed and that the confidence interval was based on a simple random sample of 15 observations. Calculate the sample mean, the margin of error, and the sample standard deviation based on the stated confidence interval and the given sample size. Use the t distribution in any calculations and round non-integer results to 4 decimal places.
Mathematics
1 answer:
vekshin13 years ago
5 0

Answer:

\bar X = \frac{2080+2260}{2}=2170

The margin of error can be founded like this:

ME = \frac{2260-2080}{2}= 90

We know that the margin of error is given by:

ME = t_{\alpha/2} \frac{s}{\sqrt{n}}

The degrees of freedom are given by:

df = n-1= 15-1 = 14

And the critical value for a 95% of confidence and 14 degrees of freedom is t_{\alpha/2} = 2.1448 then we can solve for s like this:

s= \frac{ME* \sqrt{n}}{t_{\alpha/2}}

And replacing we got:

s= \frac{90*\sqrt{15}}{2.1448}= 162.5180

Step-by-step explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

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".

\bar X represent the sample mean for the sample  

\mu population mean (variable of interest)

s represent the sample standard deviation

n represent the sample size  

Solution to the problem

The confidence interval for the mean is given by the following formula:

\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}   (1)

For this case we have the confidence interval given (2080,2260)

We can estimate the sample mean like this:

\bar X = \frac{2080+2260}{2}=2170

The margin of error can be founded like this:

ME = \frac{2260-2080}{2}= 90

We know that the margin of error is given by:

ME = t_{\alpha/2} \frac{s}{\sqrt{n}}

The degrees of freedom are given by:

df = n-1= 15-1 = 14

And the critical value for a 95% of confidence and 14 degrees of freedom is t_{\alpha/2} = 2.1448 then we can solve for s like this:

s= \frac{ME* \sqrt{n}}{t_{\alpha/2}}

And replacing we got:

s= \frac{90*\sqrt{15}}{2.1448}= 162.5180

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A population of values has a normal distribution with μ = 202.1 and σ = 24 . You intend to draw a random sample of size n = 64 .
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Answer: p= 0.0532

Step-by-step explanation: from the question

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Since the sample size is greater than 30 and the population standard deviation is known, then the z score is the appropriate to get the probabilistic value.

Z = (x - u) /(σ/√n)

Where x = sample mean.

At x = 204.5, we have the z score as

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Since the distribution is normal, we compute the probability value attached to the z score using the z table.

Graphical illustration as attachment in the answer.

Note the area under the z score is the probability value of the z score.

As we can see, the area to the left of the shaded area is a, the shaded area is b and the area to the right of the shaded area is c

We have that

a + b + c = 1

Note, the table I'm using only gives area to the left of the distribution.

Area a

To get area a, we use a z score table whose area is to the left of the distribution.

At 0.8, according to the table the area left of 0.8 is 0.78814.

Area c

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Area c = 1 - 0.84134 = 0.15866.

Thus, a = 0.78814, c = 0.15866

a + b + c = 1

0.78814 + b + 0.15866 = 1

0.9468 + b = 1

b = 1 - 0.9468

b = 0.0532.

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Answer:

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Step-by-step explanation:

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