Answer :
To find [tex]\( P(z > 0.67) \)[/tex] using the standard normal distribution, follow these steps:
1. Understand the Problem: We need to find the probability [tex]\( P(z > 0.67) \)[/tex] where [tex]\( z \)[/tex] is a value from the standard normal distribution.
2. Find the Cumulative Distribution Function (CDF): The CDF of the standard normal distribution gives us the probability that a normally distributed random variable [tex]\( Z \)[/tex] will be less than or equal to a certain value. In statistical tables or software, you can look up the value of the CDF for [tex]\( z = 0.67 \)[/tex].
3. Find the CDF Value for [tex]\( z = 0.67 \)[/tex]: By referencing standard normal distribution tables or using statistical functions, we find that the cumulative distribution function value for [tex]\( z = 0.67 \)[/tex] is approximately [tex]\( 0.7486 \)[/tex]. This means that [tex]\( P(Z \leq 0.67) = 0.7486 \)[/tex].
4. Calculate the Complement: To find [tex]\( P(Z > 0.67) \)[/tex], we need to subtract the CDF value from 1. This is because the total probability under the normal distribution curve is 1. Therefore,
[tex]\[ P(Z > 0.67) = 1 - P(Z \leq 0.67) \][/tex]
Substituting the CDF value we found:
[tex]\[ P(Z > 0.67) = 1 - 0.7486 = 0.2514 \][/tex]
5. Round the Result: The probability is already rounded to four decimal places in the previous step, which is [tex]\( 0.2514 \)[/tex].
Thus, [tex]\( P(z > 0.67) \)[/tex] using the standard normal distribution, rounded to the nearest ten-thousandths place, is [tex]\( 0.2514 \)[/tex].
1. Understand the Problem: We need to find the probability [tex]\( P(z > 0.67) \)[/tex] where [tex]\( z \)[/tex] is a value from the standard normal distribution.
2. Find the Cumulative Distribution Function (CDF): The CDF of the standard normal distribution gives us the probability that a normally distributed random variable [tex]\( Z \)[/tex] will be less than or equal to a certain value. In statistical tables or software, you can look up the value of the CDF for [tex]\( z = 0.67 \)[/tex].
3. Find the CDF Value for [tex]\( z = 0.67 \)[/tex]: By referencing standard normal distribution tables or using statistical functions, we find that the cumulative distribution function value for [tex]\( z = 0.67 \)[/tex] is approximately [tex]\( 0.7486 \)[/tex]. This means that [tex]\( P(Z \leq 0.67) = 0.7486 \)[/tex].
4. Calculate the Complement: To find [tex]\( P(Z > 0.67) \)[/tex], we need to subtract the CDF value from 1. This is because the total probability under the normal distribution curve is 1. Therefore,
[tex]\[ P(Z > 0.67) = 1 - P(Z \leq 0.67) \][/tex]
Substituting the CDF value we found:
[tex]\[ P(Z > 0.67) = 1 - 0.7486 = 0.2514 \][/tex]
5. Round the Result: The probability is already rounded to four decimal places in the previous step, which is [tex]\( 0.2514 \)[/tex].
Thus, [tex]\( P(z > 0.67) \)[/tex] using the standard normal distribution, rounded to the nearest ten-thousandths place, is [tex]\( 0.2514 \)[/tex].
