a given normal distribution with u=70 and o=6.
a- what is the probability of a randomly drawing a sample of n=9 with a mean graeter than 32?
b- what is the probability of a randomly drawing a score that falls between 20 and 35?
To calculate the probabilities in a normal distribution, you can use the Z-score formula and a standard normal distribution table.
Z-score formula:
Z = (X - μ) / σ
Where:
Z is the Z-score
X is the given value
μ is the mean
σ is the standard deviation
a) Probability of a mean greater than 32 from a sample of n=9:
To find the probability of a mean greater than 32, we need to calculate the Z-score for 32 and find the corresponding area under the normal distribution curve.
Z = (32 - 70) / 6
Z = -38 / 6
Z = -6.33
Using a standard normal distribution table or calculator, you can find that the area corresponding to a Z-score of -6.33 is very close to 0. Therefore, the probability of randomly drawing a sample of n=9 with a mean greater than 32 is approximately 0.
b) Probability of a score falling between 20 and 35:
To find the probability of a score falling between 20 and 35, we need to calculate the Z-scores for both values and find the area under the normal distribution curve between these Z-scores.
Z1 = (20 - 70) / 6
Z1 = -50 / 6
Z1 = -8.33
Z2 = (35 - 70) / 6
Z2 = -35 / 6
Z2 = -5.83
Using a standard normal distribution table or calculator, find the area corresponding to the Z-scores -8.33 and -5.83, then subtract the smaller value from the larger value to get the probability between them.
Let's say the area for Z1 is A1 and the area for Z2 is A2.
Probability = A2 - A1
Note that the values of Z1 and Z2 might be beyond the range of most standard normal distribution tables. In such cases, you can use a calculator or software that can provide more accurate probabilities for extreme Z-values.
It's important to understand that these calculations are based on the assumption that the given distribution is exactly normal. If it's an approximation or an empirical distribution, the results may vary.