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]
Are the conditions for using the normal distribution met?
Answer (2)
Check if n 1 ⋅ p ^ 1 ≥ 10 and n 1 ⋅ ( 1 − p ^ 1 ) ≥ 10 for population 1: 45 ⋅ 0.689 = 31.005 ≥ 10 and 45 ⋅ 0.311 = 13.995 ≥ 10 .
Check if n 2 ⋅ p ^ 2 ≥ 10 and n 2 ⋅ ( 1 − p ^ 2 ) ≥ 10 for population 2: 29 ⋅ 0.897 = 26.013 ≥ 10 and 29 ⋅ 0.103 = 2.987 < 10 .
The conditions are not met for population 2.
Therefore, the normal distribution cannot be used. False
Explanation
Understand the problem and provided data We are given two populations with sample sizes n 1 = 45 and n 2 = 29 , and sample proportions p ^ 1 = 0.689 and p ^ 2 = 0.897 . We need to determine if the normal distribution can be used to compare the population proportions.
State the conditions for using normal distribution To use the normal distribution for comparing population proportions, we need to check if the following conditions are met for both populations:
n 1 ⋅ p ^ 1 ≥ 10 and n 1 ⋅ ( 1 − p ^ 1 ) ≥ 10
n 2 ⋅ p ^ 2 ≥ 10 and n 2 ⋅ ( 1 − p ^ 2 ) ≥ 10
Check conditions for population 1 Let's check the conditions for population 1:
n 1 ⋅ p ^ 1 = 45 ⋅ 0.689 = 31.005 n 1 ⋅ ( 1 − p ^ 1 ) = 45 ⋅ ( 1 − 0.689 ) = 45 ⋅ 0.311 = 13.995
Since 31.005 ≥ 10 and 13.995 ≥ 10 , the conditions are met for population 1.
Check conditions for population 2 Now let's check the conditions for population 2:
n 2 ⋅ p ^ 2 = 29 ⋅ 0.897 = 26.013 n 2 ⋅ ( 1 − p ^ 2 ) = 29 ⋅ ( 1 − 0.897 ) = 29 ⋅ 0.103 = 2.987
Since 26.013 ≥ 10 but 2.987 < 10 , the conditions are not met for population 2.
Conclusion Since the conditions for using the normal distribution are not met for both populations (specifically, population 2), we cannot use the normal distribution to compare the population proportions.
Examples
In medical research, you might want to compare the proportion of patients who respond positively to two different treatments. Before using a normal approximation to perform a hypothesis test, you need to ensure that the sample sizes are large enough and the proportions are not too close to 0 or 1. This ensures the validity of the statistical test.
The normal distribution cannot be used to compare the population proportions because the conditions required for population 2 are not met. Specifically, while the first condition for population 2 is satisfied, the second condition fails as n 2 ⋅ ( 1 − p ^ 2 ) < 10 . Thus, we conclude that the normal distribution is not applicable in this case.
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