Discuss the importance of constructing confidence intervals for the population mean by answering these questions.
What are confidence intervals?
What is a point estimate?
What is the best point estimate for the population mean? Explain.
Why do we need confidence intervals?
Answer and Explanation:
Enter your step-by-step answer and explanations here.
Usually, the main question in statistical research is to estimate the essential characteristics that describe the population. Researchers use the characteristics of a sample referred to as statistics to estimate the population parameters. The population estimates are mainly calculated using point estimates or confidence intervals (Camm et al., 2018) . Point estimates are single numeric approximates such as sample mean, standard deviation, or proportion. Contrary, confidence intervals estimate the population parameters as a range of possible numbers. Confidence intervals, therefore, give a broader selection of probable values that represent population parameters. A range of possible values covers the sampling errors that cause samples from the same population to have different point estimates (Camm et al., 2018) . The confidence intervals of multiple sufficiently large random samples from the same population contain the actual population parameter. Therefore, the best estimate of the population mean is the confidence interval of the sample mean, which is more likely to contain the actual population parameter than the point estimate sample mean.
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Using the data from the Excel workbook, construct a 95% confidence interval for the population mean. Assume that your data is normally distributed and σ is unknown. Include a statement that correctly interprets the confidence interval in context of the scenario.
Hint: Use the sample mean and sample standard deviation from Deliverable 1.
Answer and Explanation:
Confidence interval of the mean when the population standard deviation is unknown calculated using the formula.
* (Camm et al., 2018).
W
here
, is the sample mean, s is the sample standard deviation, and n is the sample size
The lower limit of the mean confidence interval is
While the upper limit of the mean is
The formula uses the sample standard deviation as an estimate of the population parameter.
Sample mean, x ̅ = 71,879.3956
Sample standard deviation, s = 23,367.3602
Sample size, n = 364
Critical value t (α/2, n-1) = T.INV.2T (0.05, (364-1)) = 1.9665
Standard error, s/√n = 23,367.3602/SQRT (364) = 1,224.7825
Margin error, t (α/2, n-1) *s/√n = 1.9665 *1,224.7825 = 2,408.5600
Lower limit, X ̅ - t (α/2, n-1) * s/√n = 71,879.3956 - 2,408.5600 = 69,470.8356
Upper limit, X ̅ - t (α/2, n-1) * s/√n = 71,879.3956 + 2,408.5600 = 74,287.9556
95% confidence interval of mean is (69,470.8356, 74,287.9556)
We are therefore 95% confident that the population salary mean falls in the range (69,470.8356, 74,287.9556)
Using the data from the Excel workbook, construct a 99% confidence interval for the population mean. Assume that your data is normally distributed and σ is unknown. Include a statement that correctly interprets the confidence interval in context of the scenario.
Hint: Use the sample mean and sample standard deviation from Deliverable 1.
Answer and Explanation:
Enter your step-by-step answer and explanations here.
(Camm et al., 2018).
W here, is the sample mean, s is the sample standard deviation, and n is the sample size
The lower limit of the mean confidence interval is
While the upper limit of the mean is
The formula uses the sample standard deviation as an estimate of the population parameter.
Sample mean, x ̅ = 71,879.3956
Sample standard deviation, s = 23,367.3602
Sample size, n = 364
Critical value t (α/2, n-1) = T.INV.2T (0.01, (364-1)) = 2.5894
Standard error, s/√n = 23,367.3602/SQRT (364) = 1,224.7825
Margin error, t (α/2, n-1) *s/√n = 2.5894*1,224.7825 = 3,171.5017
Lower limit, X ̅ - t (α/2, n-1) * s/√n = 71,879.3956 - 3,171.5017= 68,707.8939
Upper limit, X ̅ - t (α/2, n-1) * s/√n = 71,879.3956 + 3,171.5017= 75,050.8973
95% confidence interval of mean is (68,707.8939, 75,050.8973)
We are therefore 99% confident that the population salary mean falls in the range (68,707.8939, 75,050.8973).
Compare your answers for (2) and (3). You notice that the 99% confidence interval is wider. What is the advantage of using a wider confidence interval? Why would you not always use the 99% confidence interval? Explain with an example.
Answer and Explanation:
Enter your step-by-step answer and explanations here.
An increase in the confidence interval means an increase in the critical t-value and hence the margin error. As the confidence interval increases, we are more confident that the population parameter will fall within a range of calculated values (Camm et al., 2018) . A 99% confidence interval may sometimes be so broad making it difficult to predict the population mean. For example, if research wishes to estimate consumers' average satisfaction rate measured in a range of 1 to 5. A 99% percent confidence interval that the mean satisfaction is between 1.5 and 4.5 provides a wide range of values. The confidence interval would not be useful in estimating the population mean.
We want to estimate the mean salary in Minnesota. How many jobs must be randomly selected for their respective mean salaries if we want 95% confidence that the sample mean is within $126 of the population mean and σ = $1150.
Is the current sample size of 364 in the data set in our Excel workbook large enough? Explain.
Answer and Explanation:
Enter your step-by-step answer and explanations here.
Sample Size, n = ( ) ^2
Where:
is the level of confidence interval
Sample size, n = ( ) ^2
= 1.96
Standard deviation, = $1150 days
Margin error, E = $126
Sample size, n = ( ) ^2 = 320.01
The minimum sample size should therefore be 320. The current sample size of 364 is therefore large enough to estimate mean salary in Minnesota.
References
Camm, J. D., Cochran, J. J., Fry, M. J., Ohlmann, J. W., and Anderson, D. R. (2018). Essentials of business analytics. Cengage Learning.
