Question

The monthly incomes from a random sample of workers in a factory are given below in dollars. Assume the population has a normal distribution and has a standard deviation of $518. Compute a 95% confidence interval for the mean of the population. Round your answers to the nearest whole dollar and use ascending order. Monthly Income 12390 12296 11916 11713 11936 11553 12000 12428 12354 12291

Answer 1

To compute a 95% confidence interval for the mean of the population, use the formula (sample mean) +/- (critical value) * (standard deviation / sqrt(sample size)).

Step-by-step explanation:

To compute a 95% confidence interval for the mean of the population, we can use the formula:

(sample mean) +/- (critical value) * (standard deviation / sqrt(sample size))

Given the sample data and the standard deviation, we can find the sample mean by taking the average of the incomes. The critical value can be found using a z-table or calculator. With a sample size of 10, the standard deviation is divided by sqrt(10). Plugging in the values, we get a 95% confidence interval of ($11627, $12460).

Answer 2

`12087.7-1.96(518)/(√(10))=11766.64`

`12087.7+1.96(518)/(√(10))=12408.76`

So on this case the 95% confidence interval would be given by (11767;12409)

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

Data: 12390 12296 11916 11713 11936 11553 12000 12428 12354 12291

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

`\bar X \pm z_(\alpha/2)(\sigma)/(√(n))` (1)

In order to calculate the mean and the sample deviation we can use the following formulas:

`\bar X= \sum_(i=1)^n (x_i)/(n)` (2)

The mean calculated for this case is
`\bar X=12087.7`

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: "=-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):

`12087.7-1.96(518)/(√(10))=11766.64`

`12087.7+1.96(518)/(√(10))=12408.76`

So on this case the 95% confidence interval would be given by (11767;12409)