Determine whether the normal distribution can be used to compare the following population proportions.
[tex]n_1=45, \quad n_2=29, \quad \hat{p}_1=0.689, \quad \hat{p}_2=0.897[/tex]
Step 1 of 2: Calculate the four values [tex]n_1 \hat{p}_1, n_1(1-\hat{p}_1), n_2 \hat{p}_2[/tex], and [tex]n_2(1-\hat{p}_2)[/tex]. Round your answers to three decimal places, if necessary.
Answer (2)
Calculate n 1 p ^ 1 = 45 × 0.689 = 31.005 .
Calculate n 1 ( 1 − p ^ 1 ) = 45 × ( 1 − 0.689 ) = 45 × 0.311 = 13.995 .
Calculate n 2 p ^ 2 = 29 × 0.897 = 26.013 .
Calculate n 2 ( 1 − p ^ 2 ) = 29 × ( 1 − 0.897 ) = 29 × 0.103 = 2.987 .
n 1 p ^ 1 = 31.005 , n 1 ( 1 − p ^ 1 ) = 13.995 , n 2 p ^ 2 = 26.013 , n 2 ( 1 − p ^ 2 ) = 2.987
Explanation
Understand the problem and provided data We are given the following data: n 1 = 45 , n 2 = 29 , p ^ 1 = 0.689 , p ^ 2 = 0.897 .
Our objective is to calculate the four values n 1 p ^ 1 , n 1 ( 1 − p ^ 1 ) , n 2 p ^ 2 , and n 2 ( 1 − p ^ 2 ) , rounding our answers to three decimal places if necessary.
Calculate n1 * p1_hat First, we calculate n 1 p ^ 1 :
n 1 p ^ 1 = 45 × 0.689 = 31.005
Calculate n1 * (1 - p1_hat) Next, we calculate n 1 ( 1 − p ^ 1 ) :
n 1 ( 1 − p ^ 1 ) = 45 × ( 1 − 0.689 ) = 45 × 0.311 = 13.995
Calculate n2 * p2_hat Now, we calculate n 2 p ^ 2 :
n 2 p ^ 2 = 29 × 0.897 = 26.013
Calculate n2 * (1 - p2_hat) Finally, we calculate n 2 ( 1 − p ^ 2 ) :
n 2 ( 1 − p ^ 2 ) = 29 × ( 1 − 0.897 ) = 29 × 0.103 = 2.987
State the final answer Therefore, the four values are: n 1 p ^ 1 = 31.005 n 1 ( 1 − p ^ 1 ) = 13.995 n 2 p ^ 2 = 26.013 n 2 ( 1 − p ^ 2 ) = 2.987
Examples
In quality control, these calculations help determine if sample sizes are large enough to reliably estimate population proportions. For instance, if a factory wants to ensure that at least 95% of its products meet certain standards, they can use these calculations to determine the sample size needed to achieve a desired level of confidence. Similarly, in political polling, these calculations are used to assess the reliability of survey results and to determine the margin of error.
We calculated four values based on the given population sizes and sample proportions: n 1 p ^ 1 = 31.005 , n 1 ( 1 − p ^ 1 ) = 13.995 , n 2 p ^ 2 = 26.013 , and n 2 ( 1 − p ^ 2 ) = 2.987 . These values are essential for determining if the normal distribution can be utilized for comparing the proportions. The calculations are rounded to three decimal places as needed.
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