Detailed Notes on Confidence Intervals for Population Mean

Confidence Interval for Population Mean ( \mu )

In statistics, a confidence interval is a range of values that is likely to contain the true population mean ( \mu ) based on a sample from the population. The interval provides an estimation of uncertainty around the sample mean.

The Empirical Rule and Standard Deviations

The Empirical Rule states that approximately 95% of the data in a normal distribution falls within 2 standard deviations of the mean. However, for more precision, especially in statistical calculations, we utilize a critical value of 1.96. This critical value corresponds to the area under the curve in a standard normal distribution and helps to define the boundaries of the confidence intervals. To derive this value:

  • First, we identify ( \alpha ), the area not between (-z) and (z): ( \alpha = 100\% - 95\% = 5\% = 0.05 ).

  • Since the distribution is symmetric, we split this area evenly between both tails, resulting in ( \alpha/2 = 0.025 ). Utilizing the standard normal table, we identify that ( P(Z < -z) = 0.025 ) yields ( z \approx 1.96 ).

Visual Explanation

This concept is further clarified through graphical representation, showing the standard normal distribution with marked areas and the corresponding z-scores.

Central Limit Theorem (CLT)

According to the Central Limit Theorem, for sufficiently large sample sizes (typically ( n \geq 30 )), the sampling distribution of the sample mean ( \bar{X} ) approaches a normal distribution with mean ( \mu ) and standard deviation ( \sigma / \sqrt{n} ). Under this theorem, we can state:

[ \text{For } 95\% \text{ of the time, } \bar{X} \text{ will be within } 1.96 \times \frac{\sigma}{\sqrt{n}} \text{ from } \mu \text{.} ]

This leads us to formulate a 95% confidence interval:
[ \bar{X} \pm 1.96 \times \frac{s}{\sqrt{n}} \text{, for sample mean } \bar{X} \text{ and sample standard deviation } s. ]

Estimating the Population Mean

When the population standard deviation ( \sigma ) is unknown, which is often the case, we substitute it with the sample standard deviation ( s ) in the confidence interval formula. However, this method carries a level of error especially if the sample size is small or if outliers exist. Hence, it is recommended to have a sample size of at least 30, and that the sample data should not exhibit outliers or significant skewness in its distribution shape.

Example: Average Number of Exclusive Relationships

Suppose a sample of 50 college students indicates a mean of 3.2 exclusive relationships with a standard deviation of 1.74. To establish a 95% confidence interval:
[
\bar{X} = 3.2, \, s = 1.74,
]

The confidence interval is computed as follows:
[
3.2 \pm 1.96 \times \frac{1.74}{\sqrt{50}} \approx 3.2 \pm 0.5 = (2.7, 3.7).
]

This interval implies we are 95% confident that the true average number of exclusive relationships among the population of college students lies between 2.7 and 3.7. However, it’s crucial to understand the distinction of statements made regarding the confidence interval:

  • True: The sample mean, denoted as ( \bar{X} ), lies within the confidence interval with 100% certainty.

  • False: It does not imply that 95% of all college students have had between 2.7 and 3.7 relationships, as that would only pertain to the population mean ( \mu ).

Confidence Intervals at Varying Confidence Levels

Confidence intervals can be constructed for various confidence levels, which influence the critical value ( z^* ). Standard confidence levels are:

  • 90% CI: ( z^* = 1.645 )

  • 95% CI: ( z^* = 1.96 )

  • 99% CI: ( z^* = 2.58 )

As the confidence level increases, the width of the confidence interval also expands, reflecting greater uncertainty about the exact value of the population mean.

Example of a 98% Confidence Interval

Given a sample size of 49 with a mean of 35 and a standard deviation of 14, we seek a 98% confidence interval:
[ \alpha = 1 - 0.98 = 0.02 \implies z_{0.01} \approx 2.326. ]

Calculating the interval:
[ 35 \pm 2.326 \times \frac{14}{\sqrt{49}} \approx 35 \pm 4.7 ]
This leads us to conclude we are 98% confident that the population mean ( \mu ) lies within ( [30.3, 39.7] ).

Small Samples and t-distribution

When the sample size is less than 30 and if the population follows a normal distribution, the t-distribution becomes applicable instead of the normal distribution. The formula for calculating the confidence interval shifts to:
[ \bar{X} \pm t{\alpha/2} \times \frac{s}{\sqrt{n}} ] Where ( t{\alpha/2} ) is obtained from the t-distribution table with ( df = n-1 ).

Example: Calculating CI with t-distribution

A random sample of 15 provides a mean of 35 with a standard deviation of 14. To calculate a 95% confidence interval:

  • With ( df = 14 ), the critical value ( t_{0.025} ) is approximately 2.145. Thus,
    [ 35 \pm 2.145 \times \frac{14}{\sqrt{15}} \approx 35 \pm 7.8 ]
    This interval is interpreted as being 95% confident the true mean ( \mu ) is within ( (27.2, 42.8) ).

Conclusion

Understanding confidence intervals is essential for making informed statistical inferences about population parameters based on sample data. Using the appropriate distribution, one can confidently report estimations that guide decision-making processes within a margin of safety and reliability. Always consider sample size and distribution characteristics while applying these statistical tools.