Question

Determine confidence intervals for each of the following:

Sample Statistic	Sample Size	Confidence Level	Confidence Interval	Lower Boundary	Upper Boundary
Mean: 150 Std. Dev.: 30	200	95%
Percent: 67%	300	99%
Mean: 5.4 Std. Dev.: 0.5	250	99%
Percent: 25.8%	500	99%

Answer 1

In the above questions,

1.)

we have to find the confidence interval for the population mean.

For this, we can use z distribution since it fulfills the conditions which are as follows:

• Ths sample size should be large usually(greater than 30 considered as large).
• We must know the population standard deviation which is also given to us.

The formula for the confidence interval is:

For this wee must know that the critical value of z which can be calculated as follows:

NOTE: We are given confidence level for 95% which can also be written as 0.95.

So, now we have to look up at the z table for an area nearest to 0.9750. so we get:

z = 1.96

Now, we have all the required value, so we just need to pot all the values in the formula, we get:

Thus, Upper bound =

Lower bound =

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2.)

Now, we have to calculate the confidence interval for given percentage which can also be said as given proportion.

We are given percentage = 67% which can be converted into proportion as

Thus, the formula for the confidence interval for a population proportion is:

where,

is the sample proportion which is given = 0.67

n is the sample size 300

and we have to find the z critical value at 99%.

So, now we have to look up at the z table for an area nearest to 0.995 which is z = 2.576

Now, we have all the required value, just need to feed in the formula we get:

Thus, Upper bound =

Lower bound =

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3.)

This question can be solved in the same way as to question 1.)

Since here also we are given population standard deviation and the sample mean

Given mean = 5.4

Standard deviation = 0.5

The formula for the confidence interval is:

For this wee must know that the critical value of z which can be calculated as follows:

NOTE: We are given confidence level for 99% which can also be written as 0.99.

So, now we have to look up at the z table for an area nearest to 0.995. so we get:

z = 2.576

Now, we have all the required value, so we just need to put all the values in the formula, we get:

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4)

This question can be solved in the same way as to question 2.)

Now, we have to calculate the confidence interval for given percentage which can also be said as given proportion.

We are given percentage = 25.8% which can be converted into proportion as

Since here also we are given population standard deviation and the sample mean

Thus, the formula for the confidence interval for a population proportion is:

where,

is the sample proportion which is given = 0.67

n is the sample size 500

and we have to find the z critical value at 99%.

So, now we have to look up at the z table for an area nearest to 0.995 which is z = 2.576

Now, we have all the required value, just need to feed in the formula we get:

Thus,

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