Lecture Notes: Statistical Analysis of M&Ms in a Bag

Overview

• The focus of this lecture is on statistical analysis to determine if the average number of M&Ms in a bag is greater than a claimed number.
• Example problem: Determine if the average number of M&Ms in a bag is more than 1000 using a sample.

Problem Setup

Hypotheses

• Null Hypothesis (H0): ( \mu = 1000 )
• Alternative Hypothesis (H1): ( \mu > 1000 )

Sample Information

• Sample size (n): 80 bags
• Sample mean (( \bar{X} )): 1036
• Sample standard deviation (S): 97.3

Statistical Analysis

Calculating Standard Error

• Standard Error (( S_{\bar{X}} )) is calculated as: [ S_{\bar{X}} = \frac{S}{\sqrt{n}} = \frac{97.3}{\sqrt{80}} = 10.8785 ]

Distribution Type

• Normally, use the T-distribution when Sigma (( \sigma )) is unknown and S is given.
• T-distribution is used due to lack of ( \sigma ).

Drawing the Distribution

• Label 0 in the middle for T-distribution.
• Mark ( \bar{X} = 1036 ) and shade to the right since ( \mu > 1000 ).

Calculating T-Score

• Formula: [ T = \frac{\bar{X} - \mu_{\bar{X}}}{S_{\bar{X}}} = \frac{1036 - 1000}{10.8785} = 3.3093 ]

Degrees of Freedom

• ( DF = n - 1 = 80 - 1 = 79 )

Finding the Area

• Use statistical software (e.g., StatDisk) to find areas:
  • Area to the right of T-score: 0.0007
  • Area to the left: 0.993

Conclusion

• 0.993 confidence that ( \mu > 1000 ).
• In English: There are more than 1000 M&Ms in a bag on average.
• ( p )-value is 0.0007; reject H0, accept H1 with 0.993 confidence.

Additional Comments

• In Excel, T-distribution only accepts positive T values.
• T or Z values are termed as "test statistics."
• Significance level ( \alpha ) is a Greek letter often resembling a fish.
• Different methods in textbooks; avoid "critical value" questions.

Next Steps

• Covered up through section 10.3; ready to work on related problems.