Determine whether the normal distribution can be used to compare the following population proportions.
[tex]n_1=44, \quad n_2=45, \quad \hat{p}_1=0.523, \quad \hat{p}_2=0.689[/tex]
Step 1 of 2 : Calculate the four values [tex]n_1 \hat{p}_1, n_1\left(1-\hat{p}_1\right), n_2 \hat{p}_2[/tex], and [tex]n_2\left(1-\hat{p}_2\right)[/tex]. Round your answers to three decimal places, if necessary.
Answer (1)
Calculate n 1 p ^ 1 = 44 × 0.523 = 23.012 .
Calculate n 1 ( 1 − p ^ 1 ) = 44 × ( 1 − 0.523 ) = 20.988 .
Calculate n 2 p ^ 2 = 45 × 0.689 = 31.005 .
Calculate n 2 ( 1 − p ^ 2 ) = 45 × ( 1 − 0.689 ) = 13.995 . Since all four values are greater than or equal to 10, the normal distribution can be used.
The normal distribution can be used: True .
Explanation
Understand the problem and provided data We are given two populations with sample sizes n 1 = 44 and n 2 = 45 . The sample proportions are p ^ 1 = 0.523 and p ^ 2 = 0.689 . We need to determine if the normal distribution can be used to compare the population proportions. To do this, we need to check if the following conditions are met:
n 1 p ^ 1 ≥ 10 n 1 ( 1 − p ^ 1 ) ≥ 10 n 2 p ^ 2 ≥ 10 n 2 ( 1 − p ^ 2 ) ≥ 10
If all four conditions are met, then the normal distribution can be used to compare the population proportions.
Calculate the required values First, we calculate n 1 p ^ 1 :
n 1 p ^ 1 = 44 × 0.523 = 23.012
Next, we calculate n 1 ( 1 − p ^ 1 ) :
n 1 ( 1 − p ^ 1 ) = 44 × ( 1 − 0.523 ) = 44 × 0.477 = 20.988
Then, we calculate n 2 p ^ 2 :
n 2 p ^ 2 = 45 × 0.689 = 31.005
Finally, we calculate n 2 ( 1 − p ^ 2 ) :
n 2 ( 1 − p ^ 2 ) = 45 × ( 1 − 0.689 ) = 45 × 0.311 = 13.995
Check the conditions and conclude Now, we check if all four conditions are met:
n 1 p ^ 1 = 23.012 ≥ 10 (True) n 1 ( 1 − p ^ 1 ) = 20.988 ≥ 10 (True) n 2 p ^ 2 = 31.005 ≥ 10 (True) n 2 ( 1 − p ^ 2 ) = 13.995 ≥ 10 (True)
Since all four conditions are met, the normal distribution can be used to compare the population proportions.
Final Answer Since all four conditions are met, the normal distribution can be used to compare the population proportions.
Examples
In quality control, you might compare the proportion of defective items from two different production lines. If the sample sizes are large enough and the proportions meet the criteria for using a normal approximation, you can use a normal distribution to test if there's a significant difference in defect rates between the two lines. This helps in identifying which production line needs improvement.
