Importance of Confidence intervals
The confidence interval is a set or range of values that a population parameter is likely to fall within (Graham, 2008). It is calculated using the sample statistics and a given confidence level. For instance, a confidence interval of the mean calculated using the sample statistics, gives the lowest and highest possible value of the population mean. The population mean is expected to be within the confidence interval.
A point estimate is a single value that approximates a population parameter (Graham, 2008). For instance, a sample mean is point estimate of the population mean, while a sample proportion is the point estimate of the population proportion. The point estimate is used in estimating the confidence interval. The confidence interval is given by the point estimate plus/minus the margin of error.
The sample mean is the most suitable point estimate of the mean of the population since it is not biased. Besides, the sample mean has the least variation from the population mean. The above features make the sample mean the most suitable point estimate of the population average.
Confidence intervals are important in statistics. In most cases, statistical analysis is based on samples since it would be expensive and time-consuming to analyze an entire population. The confidence interval helps in estimating the population parameters using the sample statistics. It also reduces the probability of errors in estimating population parameters. Using a single value for an estimate of the population parameter is likely to be inaccurate than using a range of values. Determining the exact value of a population parameter using the sample statistics is impossible
95% confidence interval
The confidence interval is given by the point estimate plus the margin of error. Determining the margin of error requires the choice of a test statistic and the confidence level. The Z-statistic is suitable for normally distributed data. The distribution usually tends to be normal if the sample size is greater than 30. In this case, salaries in Minnesota are normally distributed hence we will use the Z-statistic.
Confidence interval = X ± Z0.05/2 ×Sn,
X = sample mean
S = sample standard deviation
n = sample size.
Z0.05/2 is the Z-score at a given 0.05 alpha.
Z score for 95% confidence, Z0.05/2 = 1.96
Sample mean, X = $62,306.13
Sample standard deviation, s = 19149.21
Sample size, n = 364
Confidence interval = 62,306.13 ± 1.96 × 19,149.21364
= 62,306.13 ± 1,967.23
Upper limit = 62,306.13 + 1,967.23 = $64,273.36
Lower limit = 62,306.13 - 1,967.23 = $60,338.90
Confidence interval: ($60,338.90 < µ < $64,273.36)
The above confidence interval indicates that there is a 95% chance that the average salary for jobs is Minnesota is between $60,338.90 and $64,273.36.
99% confidence interval
Confidence interval = X ± Z0.01/2 ×Sn
Z score for 99% confidence, Z0.01/2 = 2.58
Sample mean, X = $62,306.13
Sample standard deviation, s = 19149.21
Sample size, n = 364
Confidence interval = 62,306.13 ± 2.58 × 19,149.21364
= 62,306.13 ± 2,589.52
Upper limit = 62,306.13 + 2,589.52 = $64,895.65
Lower limit = 62,306.13 - 2,589.52 = $59,717.13
Confidence interval: ($59,717.13< µ < $64,895.65)
The above confidence interval indicates that there is a 95% chance that the average salary for jobs is Minnesota is between $59,717.13 and $64,895.65.
Comparing 95% and 99% confidence intervals
As shown above, the 95% confidence interval is narrower than the 99% confidence interval. Thus, the confidence interval is wider for a higher confidence level than for a lower confidence level. If a higher confidence level is required, the confidence interval must be wider.
References
Graham, A. (2008). Statistics (1st ed.). Blacklick, OH: McGraw-Hill.
