The monthly incomes from a random sample of workers in a factory are shown
below.
Monthly Income
(In $1,000)
4.0
5.0
7.0
4.0
6.0
6.0
7.0
9.0
a. Compute the standard error of the mean (in dollars).
b. Compute the margin of error (in dollars) at 95% confidence.
c. Compute a 95% confidence interval for the mean of the population.
Assume the population has a normal distribution. Give your answer in dollars.
You will need to compute the mean and standard deviation. Once you have those values, you can compute the rest using formulas.
a) standard error of the mean = sd/√n
Note: sd = standard deviation; n = sample size
b) Margin of error = 1.96(sd/√n)
Note: 1.96 represents 95% using a z-table
c) CI95 = mean ± 1.96(sd/√n)
I'll let you take it from here.
To answer these questions, we need to calculate the standard error of the mean, the margin of error, and the confidence interval.
a. To compute the standard error of the mean, we use the formula:
Standard Error of the Mean = Standard Deviation / √(Sample Size)
First, let's calculate the sample size, which is the number of monthly incomes in the sample, given as 8.
Next, we need to calculate the standard deviation, which measures the spread of the data around the mean. To do this, we follow these steps:
1. Find the mean (average) of the monthly incomes:
Mean = (sum of monthly incomes) / (sample size)
Mean = (4.0 + 5.0 + 7.0 + 4.0 + 6.0 + 6.0 + 7.0 + 9.0) / 8 = 6.125
2. Calculate the differences between each monthly income and the mean:
Differences = monthly income - mean
The differences for each monthly income are:
0.875, -1.125, 0.875, -2.125, -0.125, -0.125, 0.875, 2.875
3. Square each difference:
Squared Differences = Difference^2
The squared differences for each monthly income are:
0.765625, 1.265625, 0.765625, 4.515625, 0.015625, 0.015625, 0.765625, 8.265625
4. Calculate the sum of the squared differences:
Sum of Squared Differences = sum of squared differences
Sum of Squared Differences = 0.015625 + 0.015625 + 0.765625 + 0.765625 + 0.765625 + 1.265625 + 4.515625 + 8.265625 = 16.375
5. Calculate the variance:
Variance = Sum of Squared Differences / (Sample Size - 1)
Variance = 16.375 / (8 - 1) = 2.339285714
6. Calculate the standard deviation:
Standard Deviation = √Variance
Standard Deviation = √2.339285714 = 1.528462156
Now, we can calculate the standard error of the mean:
Standard Error of the Mean = Standard Deviation / √(Sample Size)
Standard Error of the Mean = 1.528462156 / √8 = 0.540832691
Therefore, the standard error of the mean is approximately $0.541 (rounded to three decimal places).
b. To compute the margin of error at 95% confidence, we multiply the standard error of the mean by the critical value. The critical value depends on the desired confidence level.
For a 95% confidence level, the critical value is approximately 1.96 (Z-value from the standard normal table).
Margin of Error = Critical Value * Standard Error of the Mean
Margin of Error = 1.96 * 0.541 = 1.06036
Therefore, the margin of error is approximately $1.060 (rounded to three decimal places).
c. To compute a 95% confidence interval for the mean of the population, we use the formula:
Confidence Interval = Mean ± (Margin of Error)
Given the mean we calculated in part a:
Mean = 6.125
And the margin of error we calculated in part b:
Margin of Error = $1.060
Confidence Interval = 6.125 ± 1.060
Lower bound of the confidence interval = Mean - Margin of Error
Lower bound = 6.125 - 1.060 = 5.065
Upper bound of the confidence interval = Mean + Margin of Error
Upper bound = 6.125 + 1.060 = 7.185
Thus, the 95% confidence interval for the mean of the population is approximately $5.065 to $7.185.
To calculate the standard error of the mean, follow these steps:
a. Compute the standard error of the mean (in dollars).
1. Calculate the sample mean, which is the sum of all incomes divided by the total number of incomes:
Sample mean = (4.0 + 5.0 + 7.0 + 4.0 + 6.0 + 6.0 + 7.0 + 9.0) / 8 = 48 / 8 = 6.
2. Calculate the standard deviation of the sample, which measures the spread of the data points around the sample mean:
1. Calculate the sum of the squared differences between each data point and the sample mean:
(4.0 - 6)^2 + (5.0 - 6)^2 + (7.0 - 6)^2 + (4.0 - 6)^2 + (6.0 - 6)^2 + (6.0 - 6)^2 + (7.0 - 6)^2 + (9.0 - 6)^2
= 2^2 + (-1)^2 + 1^2 + 2^2 + 0^2 + 0^2 + 1^2 + 3^2
= 4 + 1 + 1 + 4 + 0 + 0 + 1 + 9
= 20.
2. Divide the sum above by n-1, where n is the sample size:
Standard deviation = sqrt(20 / (8 - 1))
= sqrt(20 / 7)
≈ sqrt(2.857)
≈ 1.69.
3. Calculate the standard error of the mean:
Standard error of the mean = standard deviation / sqrt(n)
= 1.69 / sqrt(8)
≈ 0.598.
Therefore, the standard error of the mean is approximately $0.598.
b. Compute the margin of error (in dollars) at 95% confidence.
To calculate the margin of error, we need to use the t-distribution. Since the sample size is small (n < 30), the t-distribution is appropriate:
1. Determine the degree of freedom, which is n - 1:
Degrees of freedom = 8 - 1 = 7.
2. Find the t-critical value for a 95% confidence level and 7 degrees of freedom. This value can be obtained from a t-table or by using statistical software. For this example, let's assume the t-critical value is 2.365.
3. Calculate the margin of error:
Margin of error = t-critical value * standard error of the mean
= 2.365 * 0.598
≈ 1.412.
Therefore, the margin of error at 95% confidence is approximately $1.412.
c. Compute a 95% confidence interval for the mean of the population.
To calculate the confidence interval, use the formula:
Confidence interval = Sample mean ± Margin of error
Plugging in the values:
Confidence interval = 6 ± 1.412
= (6 - 1.412, 6 + 1.412)
= (4.588, 7.412)
Therefore, the 95% confidence interval for the mean of the population is approximately $4.588 to $7.412.