Homework Chapter 3.2 Notes

Homework Chapter 3.2 Analysis

Context and Objective

  • This homework assigned analyzes a dataset representing the downloads of a new mobile app from the app store on its launch day.
  • The data is measured in thousands.
  • The primary goal is to find the sample variance and the standard deviation for the provided sample of download counts.

Dataset

  • The dataset includes the following download figures (in thousands):
    • 15
    • 36
    • 20
    • 16
    • 13
    • 14

Concepts Defined

Sample Variance
  • Definition: The sample variance ( ext{s}^2) measures the dispersion of a sample. It is calculated as the average of the squared deviations from the sample mean.
  • Formula:
    ext{s}^2 = rac{1}{n-1} imes ext{Sum of squared deviations from the mean}
    where (n) is the number of observations in the sample.
Standard Deviation
  • Definition: The standard deviation (s) is the positive square root of the variance, providing a measure of the average distance of the data points from the mean.
  • Formula:
    s=extsqrt(s2)s = ext{sqrt}(s^2)

Calculation Steps

Tools Needed
  • Calculator: The TI-84 Plus calculator will be utilized for computations based on the dataset.
Step 1: Determine the Sample Mean ((\bar{x}))
  • Calculate the Mean:
    xˉ=Sum of all observationsn\bar{x} = \frac{\text{Sum of all observations}}{n}
    xˉ=15+36+20+16+13+146=1146=19\bar{x} = \frac{15 + 36 + 20 + 16 + 13 + 14}{6} = \frac{114}{6} = 19
Step 2: Calculate Each Deviation from the Mean
  • Deviations:
    • 15 - 19 = -4
    • 36 - 19 = 17
    • 20 - 19 = 1
    • 16 - 19 = -3
    • 13 - 19 = -6
    • 14 - 19 = -5
Step 3: Square Each Deviation
  • Squared Deviations:
    • (-4)^2 = 16
    • (17)^2 = 289
    • (1)^2 = 1
    • (-3)^2 = 9
    • (-6)^2 = 36
    • (-5)^2 = 25
Step 4: Calculate the Sum of Squared Deviations
  • Sum:
    16+289+1+9+36+25=37616 + 289 + 1 + 9 + 36 + 25 = 376
Step 5: Calculate Sample Variance ((\text{s}^2))
  • Using Formula:
    s2=376n−1=3765=75.20\text{s}^2 = \frac{376}{n - 1} = \frac{376}{5} = 75.20
Step 6: Calculate Standard Deviation (s)
  • Standard Deviation Calculation:
    s=75.20=8.67s = \sqrt{75.20} = 8.67

Conclusion

  • The sample variance for the download times is 75.20.
  • The sample standard deviation for the download times is 8.67.

Final Notes

  • Ensure all values are rounded to two decimal places where applicable.
  • This illustrates how variability can be quantified in data samples, useful for determining the spread of app downloads.