Importance of constructing confidence intervals for the population mean
Confidence interval is a range of figures that provides an interval estimate of a set of unknown parameters (Heckard, Utts, & Utts, 2012). This is as opposed to using point estimation and contains the parameter’s value as well as stated probability.
Point estimate, on the other hand, uses a set of sample data to calculate a statistic (a single value) which serves as the best estimate of unknown parameter in a population whether random or fixed (Heckard, Utts, & Utts, 2012).
The best point for the population mean, E(X), is the sample mean, Xbar. By equating the population mean with the sample mean, we are solving for the parameters using the one-parameter case.
Confidence interval (C.I) is needed for bounding the mean and the standard deviation. In addition, the C.I will also be needed for obtaining the proportions, regression coefficients and the differences for the population proportions (Heckard, Utts, & Utts, 2012). C.I is also needed in obtaining and estimating the sampling error in relation to the parameter of interest.
Best point estimate for the population mean
The mean is the average of the data set and normally the centre of the data.
Sample Mean = Total of Ages / Sample Size
Sample Mean = 3705 / 60 = 61.81667
Sample Mean () = 61.82
The sample mean is () is the best point estimate of the population mean (µ).
The best point estimate for the population mean (µ) = 61.82
Confidence intervals for the population mean
Assuming that your data is normally distributed and the population standard deviation is unknown:
At 95% confident level:
C.I is given by:
With = 61.82, n = 60, s = 8.84597, n-1 = 59
Margin of error =
C.I. = 61.82 1.9083 = (59.9117, 63.7283)
At 99% confident level:
With = 61.82, n = 60, s = 8.84597, n-1 = 59
Margin of error =
C.I. = 61.82 3.0446 = (58.7754, 64.8646)
From the computations above, it can be seen that at 95% confidence level, the interval of the population mean lies between 59.9117 and 63.7283. The sample mean is 61.82 and therefore the mean lies within the interval of the figures. After increasing the confidence level to 99%, the interval also increases. At 99% confidence level, the sample mean is still within the interval as range of the interval figures is 58.7754 and 64.8646.
There are a number of observations that can be made by changing the confidence levels from 95% to 99%. First, the margin of error increase from 1.9083 to 3.8646. The critical value read off from the t-table also increases. The confidence interval also widens as a result of increasing the confidence interval form 95% to 99%. If the degree of confidence is increased, it affects the margin of error. The larger the confidence degree (level), the larger the margin of error (Heckard, Utts, & Utts, 2012). Since the confidence level is being increased from 95% to 99%, then the margin of error is also increased. This ultimately increases the confidence interval of the population mean.
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