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Menso is an organisation for people with high Intelligence Quotients (IQs). Menso is investigating the average IQ of primary school students to determine whether its entry requirements should be altered for younger people. A sample of 50 primary school students have been randomly selected from schools throughout the country. The sample mean IQ of those students was calculated as 100. It is known that the population standard deviation of the IQs of all people is 10. It is assumed that this standard deviation will also apply specifically to the IQs of the primary school students. Calculate the upper and lower bounds of the 95% confidence interval for the mean IQ of primary school students. Give your answers to 2 decimal places.

Respuesta :

Answer:

Lower bound: 97.23

Upper bound: 102.77

Step-by-step explanation:

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

[tex]\alpha = \frac{1-0.95}{2} = 0.025[/tex]

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

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

Now, find M as such

[tex]M = z*\frac{\sigma}{\sqrt{n}}[/tex]

In which [tex]\sigma[/tex] is the standard deviation of the population and n is the size of the sample.

[tex]M = 1.96*\frac{10}{\sqrt{50}} = 2.77[/tex]

The lower end of the interval is the mean subtracted by M. So it is 100 - 2.77 = 97.23

The upper end of the interval is the mean added to M. So it is 100 + 2.77 = 102.77

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