Statistics Chapter 8: Effect of Sample Size on Standard Error of the Mean

Effect of Sample Size on Standard Error of the Mean

Overview of Typical Measurements

1. Population Description
   • Focused on a distribution of American males who are all five foot nine inches tall.
   • Population Mean Weight: 160 pounds
   • Population Standard Deviation: 10 pounds

Core Concepts

1. Population Statistics

   • Population Mean: The average weight of the population is given as ext{Population Mean} = 160 ext{ pounds}.
   • Population Standard Deviation: The variability of weights around the mean is described with a standard deviation of ext{Population Standard Deviation} = 10 ext{ pounds}.

2. Sample Size and Standard Error

   • The relationship between sample size and standard error is crucial for understanding variability in sample means.
   • The Standard Error of the Mean (SEM) is calculated to determine how much sampled means are expected to vary from the population mean.

Calculation of Standard Error for Different Sample Sizes

1. Sample Sizes Considered: 36, 100, 400, and 1600
2. Mean of the Sampling Distribution: Regardless of sample size, the mean of the sampling distribution remains the same as the population mean:
   ext{Mean of Sampling Distribution} = 160 ext{ pounds}

Steps for Calculating Standard Error

1. For Sample Size = 36:

   • Calculate:
     ext{Standard Error} = rac{10}{ ext{sqrt}(36)}
   • Computation:
     ext{Standard Error} = rac{10}{6} = 1.6667

2. For Sample Size = 100:

   • Calculate:
     ext{Standard Error} = rac{10}{ ext{sqrt}(100)}
   • Computation:
     ext{Standard Error} = rac{10}{10} = 1
   • Observation: Standard error decreased from 1.6667 to 1 as sample size increased.

3. For Sample Size = 400:

   • Calculate:
     ext{Standard Error} = rac{10}{ ext{sqrt}(400)}
   • Computation:
     ext{Standard Error} = rac{10}{20} = 0.5
   • Continuing Observation: Standard error decreased further to 0.5.

4. For Sample Size = 1600:

   • Calculate:
     ext{Standard Error} = rac{10}{ ext{sqrt}(1600)}
   • Computation:
     ext{Standard Error} = rac{10}{40} = 0.25
   • Final Observation: Standard error decreases further to 0.25.

Implications of Changing Sample Size on Standard Error

1. Compiling Results:

   • The relationship established shows that increasing the sample size leads to a direct decrease in the standard error:
     • Sample Size 36: Standard Error = 1.6667
     • Sample Size 100: Standard Error = 1.0
     • Sample Size 400: Standard Error = 0.5
     • Sample Size 1600: Standard Error = 0.25

2. Interpretation of Standard Error:

   • A smaller standard error indicates that the sample means are less dispersed and more clustered about the population mean.
   • In contrast, a larger standard error suggests more variability in the sampled means.

Visual Representation of Data Distribution Changes

1. Distribution Shape:
   • As sample size increases from 36 to 1600, the mean of the mass of sample means clusters more tightly around the population mean.
   • Visualization involves peaks of bell curves; with larger sample sizes, the peak rises indicating more data values near the mean.

Practical Application of Sample Size Advantages

1. Confidence in Data Representation:
   • Increasing sample size results in decreased standard error which enhances the reliability of sample means as representational of the population mean.
2. Range Calculation:
   • The range is calculated using standard error expansions.
   • For example:
     • If using 2.58 standard errors, calculate lower and upper bounds as follows:
     • For n = 1600:
     • Lower Bound: 160 - 2.58 imes 0.25 = 159.35
     • Upper Bound: 160 + 2.58 imes 0.25 = 160.65
   • The range between low and high means:
     • Smaller ranges indicate better confidence in a mean that correctly represents the population mean.

Conclusory Insights

1. Optimal Sample Size:
   • To maximize the likelihood of selecting a representative sample mean, increase sample size to lower standard error effectively.
   • Larger sample sizes yield less variability and increased homogeneity around the population mean, enhancing predictive accuracy.