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Elena L [17]
3 years ago
9

A random sample of n measurements was selected from a population with unknown mean μ and known standard deviation σ. Calculate a

95% confidence interval for μ for the given situation. Round to the nearest hundredth when necessary. n = 125, x-bar = 91, σ = 20 Group of answer choices
Mathematics
2 answers:
vivado [14]3 years ago
5 0

Answer:

The 95% confidence interval for μ for the given situation is between 87.49 and 94.51.

Step-by-step explanation:

We have that to find our \alpha level, that is the subtraction of 1 by the confidence interval divided by 2. So:

\alpha = \frac{1-0.95}{2} = 0.025

Now, we have to find z in the Ztable as such z has a pvalue of 1-\alpha.

So it is z with a pvalue of 1-0.025 = 0.975, so z = 1.96

Now, find the margin of error M as such

M = z*\frac{\sigma}{\sqrt{n}}

In which \sigma is the standard deviation of the population and n is the size of the sample.

M = 1.96\frac{20}{\sqrt{125}} = 3.51

The lower end of the interval is the sample mean subtracted by M. So it is 91 - 3.51 = 87.49

The upper end of the interval is the sample mean added to M. So it is 91 + 3.51 = 94.51

The 95% confidence interval for μ for the given situation is between 87.49 and 94.51.

Zanzabum3 years ago
3 0

Answer:

91-1.96\frac{20}{\sqrt{125}}=87.494    

91+1.96\frac{20}{\sqrt{125}}=94.506    

So on this case the 95% confidence interval would be given by (87.494; 94.506). We have 95% of confidence that the true mean is between the limits founded.  

Step-by-step explanation:

Notation

\bar X=91 represent the sample mean

\mu population mean (variable of interest)

\sigma=20 represent the population standard deviation

n=125 represent the sample size  

Confidence interval

The confidence interval for the mean if we know the deviation is given by the following formula:

\bar X \pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}   (1)

Since the Confidence is 0.95 or 95%, the value of significance is \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: "=-NORM.INV(0.025,0,1)".And we see that z_{\alpha/2}=1.96

Now we have everything in order to replace into formula (1):

91-1.96\frac{20}{\sqrt{125}}=87.494    

91+1.96\frac{20}{\sqrt{125}}=94.506    

So on this case the 95% confidence interval would be given by (87.494; 94.506). We have 95% of confidence that the true mean is between the limits founded.    

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