7.012

AP Statistics: Module Seven: Means and Slope

07.01: Hypothesis Testing—One-Sample Mean

Introduction

  • Context: A manufacturer of the Kiddie-Go-Round requires that each seat's mean weight limit should not exceed 110 pounds.

  • Objective: To test if the manufacturer's seats meet this requirement using a sample.

Hypothesis Test

Parameters

  • Let ( \mu ) equal the true mean weight limit of a Kiddie-Go-Round seat.

  • Null Hypothesis (H0): ( \mu = 110 ) (The mean weight limit of a Kiddie-Go-Round seat is 110 pounds.)

  • Alternative Hypothesis (Ha): ( \mu > 110 ) (The mean weight limit of a Kiddie-Go-Round seat is greater than 110 pounds.)

Conditions for Hypothesis Testing

  • Simple Random Sample: Sample obtained was chosen at random.

  • Independence: Assumption holds as there are more than 250 Kiddie-Go-Round seats, ensuring independence.

  • Normality: Sample size ( n = 25 ), checking for large outliers or extreme skew. Caution should be taken if conditions are violated.

Page 2: Performing a One-Sample t-Test

Calculations

  • Significance Level: ( \alpha = 0.05 )

  • t-Statistic Calculation:[ t = \frac{112.7 - 110}{9.62 / \sqrt{25}} = \frac{2.7}{1.924} \approx 1.4033 ]

  • Finding p-value:

    • Use Table B in the AP Resource Packet with ( df = 25 - 1 = 24 )

  • t Distribution Critical Values: Locate where ( t = 1.4033 ) falls.

Critical Values from Table B

  • t-values for different tail probabilities up to 0.10 and 0.05 are read off:

    • 1.318: p = 0.05

    • 1.729: p = 0.10

Page 3: Conclusion from t-Test

p-Value Analysis

  • Finding p-value Range: ( t = 1.4033 ) lies between 1.318 and 1.729, indicating:

  • p-value: Between 0.05 and 0.10.

  • Exact p-value Calculation: Use calculator to find exact p-value using the command tcdf.

Result of Statistical Test

  • Conclusion: Calculated p-value ( p = 0.0867 ) is greater than ( \alpha = 0.05 )

  • Decision: Fail to reject the null hypothesis.

  • Interpretation: Not sufficient evidence to suggest that the mean weight limit of a Kiddie-Go-Round seat exceeds 110 pounds.

Page 4: Significance Level Impact

Changing Significance Level to 0.10

  • Comparison: If ( \alpha = 0.10 ):

    • Since ( p = 0.0867 < 0.10 ), reject H0.

  • Conclusion Change: Sufficient evidence suggests the mean weight limit is greater than 110 pounds.

Key Takeaway

  • Different significance levels can lead to different conclusions regarding hypothesis testing, illustrating the inherent variation in statistics.

  • Type I Error Consideration: Selecting a level of significance affects the chance of making a Type I error.