Anonymous
May 27
(b) The probability that the sample proportion of population 2 exceeds that of population 1 by 7%.?
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ChatGPT
May 27
To find the probability that the sample proportion of population 2 exceeds that of population 1 by 7%, we can use the normal distribution and the formula for the difference between two sample proportions.
Let p1 be the sample proportion of population 1 and p2 be the sample proportion of population 2. We want to find P(p2 - p1 > 0.07).
The standard error of the difference between two sample proportions is given by:
SE = sqrt(p1(1-p1)/n1 + p2(1-p2)/n2)
Assuming that the null hypothesis is true (p2 = p1), we can calculate the standard error using the pooled sample proportion:
p = (n1*p1 + n2*p2) / (n1 + n2)
Then, the standard error becomes:
SE = sqrt(p*(1-p)*(1/n1 + 1/n2))
Now, we can calculate the z-score for the difference of 7%:
z = (0.07) / SE
Finally, we can find the probability that the sample proportion of population 2 exceeds that of population 1 by 7% by looking up the z-score in a standard normal distribution table or using a statistical software.
Let p1 be the sample proportion of population 1 and p2 be the sample proportion of population 2. We want to find P(p2 - p1 > 0.07).
The standard error of the difference between two sample proportions is given by:
SE = sqrt(p1(1-p1)/n1 + p2(1-p2)/n2)
Assuming that the null hypothesis is true (p2 = p1), we can calculate the standard error using the pooled sample proportion:
p = (n1*p1 + n2*p2) / (n1 + n2)
Then, the standard error becomes:
SE = sqrt(p*(1-p)*(1/n1 + 1/n2))
Now, we can calculate the z-score for the difference of 7%:
z = (0.07) / SE
Finally, we can find the probability that the sample proportion of population 2 exceeds that of population 1 by 7% by looking up the z-score in a standard normal distribution table or using a statistical software.
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Google Gemini AI
May 27
We are given that the sample size for population 1 is n1 = 500 and the sample proportion for population 1 is p1 = 0.6. The sample size for population 2 is n2 = 400 and the sample proportion for population 2 is p2 = 0.67. We want to find the probability that p2 - p1 > 0.07.
We can use the central limit theorem to approximate the distribution of p2 - p1. The mean of the distribution is:
E(p2 - p1) = E(p2) - E(p1) = 0.67 - 0.6 = 0.07
The variance of the distribution is:
Var(p2 - p1) = Var(p2) + Var(p1) = (0.67 - 0.33) / 400 + (0.6 - 0.4) / 500 = 0.000441
The standard deviation of the distribution is:
SD(p2 - p1) = sqrt(Var(p2 - p1)) = 0.021
We can now standardize the distribution:
Z = (p2 - p1 - E(p2 - p1)) / SD(p2 - p1)
We want to find the probability that Z > 0.07 / 0.021 = 3.33. We can use a standard normal table or a calculator to find that the probability is:
P(Z > 3.33) = 1 - P(Z < 3.33) = 1 - 0.9995 = 0.0005
Therefore, the probability that the sample proportion of population 2 exceeds that of population 1 by 7% is 0.0005.
We can use the central limit theorem to approximate the distribution of p2 - p1. The mean of the distribution is:
E(p2 - p1) = E(p2) - E(p1) = 0.67 - 0.6 = 0.07
The variance of the distribution is:
Var(p2 - p1) = Var(p2) + Var(p1) = (0.67 - 0.33) / 400 + (0.6 - 0.4) / 500 = 0.000441
The standard deviation of the distribution is:
SD(p2 - p1) = sqrt(Var(p2 - p1)) = 0.021
We can now standardize the distribution:
Z = (p2 - p1 - E(p2 - p1)) / SD(p2 - p1)
We want to find the probability that Z > 0.07 / 0.021 = 3.33. We can use a standard normal table or a calculator to find that the probability is:
P(Z > 3.33) = 1 - P(Z < 3.33) = 1 - 0.9995 = 0.0005
Therefore, the probability that the sample proportion of population 2 exceeds that of population 1 by 7% is 0.0005.
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