A random variable is normally distributed. It has a mean of 245 and a standard d

Question

A random variable is normally distributed. It has a mean of 245 and a standard deviation of 21 a.) If you take a sample of size 10, can you say what the shape of the distribution b.) For a sample of size 10, state the mean of the sample mean and the standard c.) For a sample of size 10, find the probability that the sample mean is more than d.) If you take a sample of size 35, can you say what the shape of the distribution e.) For a sample of size 35, state the mean of the sample mean and the standard f.) For a sample of size 35, find the probability that the sample mean is more than g.) Compare your answers in part d and f. Why is one smaller than the other? for the sample mean is? Why? deviation of the sample mean. 241 of the sample mean is? Why? deviation of the sample mean. 241

Explanation / Answer

a) we know that When the sample size is sufficiently large, the shape of the sampling distribution approximates a normal curve

however in this case the sample size is not large enough , hence the shape can be anything but not normal

b)
using the CLT , the mean would be 245
and the standard deviation would be
sd/sqrt(n)

= 21/sqrt(10) = 6.64

c)

P(x>241)

we knwo that the z score is given as

z =
(x-mean)/(sd/sqrt(n)) = (241-245)/(21/sqrt(10)) = -0.602

please keep the z tables ready

P ( Z>0.602 )=P ( Z<0.602 )=0.7257

d)

if the sample size is increased from 10 to 35 , then the shape of the distribution would be more towards the normal distribution (as that of population)

Please note that we can answer only 4 subparts of a question at a time , as per the answering guidelines

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