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.

If you take a sample of size 10, can you say what the shape of the distribution for the sample mean is? Why?

For a sample of size 10, state the mean of the sample mean and the standard deviation of the sample mean.

For a sample of size 10, find the probability that the sample mean is more than 241.

If you take a sample of size 35, can you say what the shape of the distribution of the sample mean is? Why?

For a sample of size 35, state the mean of the sample mean and the standard deviation of the sample mean.

For a sample of size 35, find the probability that the sample mean is more than 241.

Compare your answers in part c and f. Why is one smaller than the other?

Explanation / Answer

Ans:

Normal distribution(bell shaped),as according to central limit theorm,when we take a sample(irrespective of sample size) from normal distribuiion,sampling distribution of sample means follow normal distribution.

when n=10

mean=245

standard deviation=21/sqrt(10)=6.64

z=(241-245)/(21/sqrt(10)

z=-0.60

P(z>-0.6)=0.7265

when n=35,shape of distribuion will be bell shaped or normal distribution.

mean=245

standard deviation=21/sqrt(35)=3.55

z=(241-245)/(21/sqrt(35))

z=-1.13

P(z>-1.13)=0.8701

Probability for n=35 is larger,as standard error decreases for larger sample size.

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