SOLUTION: The weights (in kg) of the students in a school is normally distributed with standard deviation 15. (a) If we know that the population mean is 67. Find the probability that the

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Question 1149925: The weights (in kg) of the students in a school is normally distributed with standard deviation 15.
(a) If we know that the population mean is 67. Find the probability that the mean weight of a random sample of 31 students is over 70kg.
(b) Assume that the population mean is unknown and we want to estimate it by taking a random sample of 31 students. It is found that the mean of the weights in the sample is 65kg. Find an approximate 90% confidence interval for the mean of the weights of students in the school.
Answer by VFBundy(438) About Me  (Show Source):
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(a) If we know that the population mean is 67. Find the probability that the mean weight of a random sample of 31 students is over 70 kg.

SD (standard deviation) of the sample = 15%2Fsqrt%2831%29 = 2.6941

%2870+-+Mean%29%2FSD%28sample%29 = %2870+-+67%29%2F2.6941 = 3%2F2.6941 = 1.11

Look up the z-score of 1.11 on a z-table. The result is 0.8665. This is the probability the mean weight of the sample is BELOW 70 kg. The question asks what the probability the mean weight of the sample is ABOVE 70 kg. Therefore, we subtract 0.8665 from 1. This gives us our answer of 0.1335.

(b) Assume that the population mean is unknown and we want to estimate it by taking a random sample of 31 students. It is found that the mean of the weights in the sample is 65 kg. Find an approximate 90% confidence interval for the mean of the weights of students in the school.

From the first part of the question, we know that the SD of the sample is 2.6941.

To find a 90% confidence interval, we want to find the z-score where the area under the curve is between 95% and 5%. So, we want to find the z-scores that correspond to the values 0.9500 and 0.0500. These scores would be +1.645 and -1.645.

So, we know the mean of the sample is 65 kg. We know the SD of the sample is 2.6941. Therefore, we can figure of the confidence interval as such:

65 + 1.645(2.6941) = 69.43
65 - 1.645(2.6941) = 60.57

So, we are 95% confident that the mean weight of students in the school is between 60.57 kg and 69.43 kg.