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IT 223 -- Apr 24, 2024

Review Exercises

1. Usually the population mean is unknown. How do we get an approximate idea of its value?
2. What are two methods for estimating the accuracy of the sample mean?
   Answer: Method 1, which is to repeat the experiment k times giving k batches of data. Then compute the standard deviation of the sample means obtained from all of the batches. Method 2 is only conduct one experiment and estimate SE_ave as SD+ / √n. We will prefer Method 2 in IT 223.
3. The Nielsen Company wants to estimate the average number of hours x per month that U.S. college students cell phone spend watching videos on their phones. They collect a random sample of size 64 and interview them. The results are x = 3.5 hours per month with SD+ - 4.0. Compute a 95% confidence interval for the true average time μ that college students spend watching videos on their phones. According to the Central Limit Theorem, even though x may not be normally distributed, x is if n is large enough. "Large enough" is usually taken to be about n = 30. Answer:
   If z is a standard normal random observation, it will be in the interval [-2, 2] 95% of the time. This is because area[-2, 2] = 0.95 = 95% (which you memorized). Now compute the z-score the sample averate:
   z = (x - μ) / SE_x. A 95% confidence interval for μ is
   -2 ≤ z ≤ 2
   -2 ≤ (x - μ) / SE_x ≤ 2
   Now x = 3.5 and SD+ = 4.0, so SE_x = SD+ / √n = 4.0 / √64 = 4 / 8 = 0.5. Therefore our inequalty becomes
   -2 ≤ (3.5 - μ) / 0.5 ≤ 2
   0.5 × (-2) ≤ 3.5 - μ ≤ 0.5 × 2
   -1 ≤ 3.5 - μ ≤ 1
   -1 - 3.5 ≤ -μ ≤ 1 - 3.5
   -4.5 ≤ -μ ≤ -2.5
   4.5 ≥ μ ≥ 2.5
   2.5 < μ < 4.5
   This means that [2.5, 4.5] is a 95% confidence interval for μ.
4. Who first stated the Central Limit Theorem?
   Answer: Abraham deMoivre.

Bivariate Datasets

• Practice Problem: Look at the Bears 2024 Roster: bears-2024-roster.txt. Use R to plot the weight in kilos vs. the height in meters for each player. The conversion rates are 0.3048 meters per foot and 0.4536 kilos per pound.

Correlation

• Practice Problem: With the Bears Roster 2024 Dataset, use R to compute the correlation of the height in meters vs. the weight in meters for each player. The conversion rates are 0.3048 meters per foot and 0.4536 kilos per pound.

Linear Regression

• Practice Problem: With the Bears Roster 2024 Dataset, use R to:
  
  1. Compute the height in meters and the weight in kilos of the Bears players. The conversion rates are 0.3048 meters per foot and 0.4536 kilos per pound.
  2. Find the simple linear regression equation for predicting weight in kilos from height in meters.
  3. Create the residual plot, which is the residuals (y-axis) vs. the predicted values (x-axis)

Project 3

• Look at the project descriptions for Project 3.