Mathematics: A sample of 10 chunks of coal from a particular mine found a sample mean density of 1.39 g/m3 with a population standard deviation of 0.78 g/m3. We wish to create a confidence interval for the true mean density of coal from this particular mine. Round all answers to three decimal places.

A sample of 10 chunks of coal from a particular mine

Ответ:

lelen2021

22.01.2022

Mathematics

$1.39-2.262\frac{0.78}{\sqrt{10}}=0.832$

$1.39+2.262\frac{0.78}{\sqrt{10}}=1.948$

Step-by-step explanation:

Assuming the following question: Compute a 95% confidence interval using the 2-distribution, as the population standard deviation is known. The bounds of this confidence interval (lower upper) are:

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".

$\bar X =1.39$ represent the sample mean

$\mu$ population mean (variable of interest)

s=0.78 represent the sample standard deviation

n=10 represent the sample size

Calculate the confidence interval

Since the sample size is large enough n<30. The confidence interval for the mean is given by the following formula:

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

The degrees of freedom are given by:

df =n-1= 10-1=9

Since the Confidence is 0.95 or 95%, the value of $\alpha=0.05$ and $\alpha/2 =0.025$ , and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.025,9)".And we see that $t_{\alpha/2}=2.262$