2) If you roll 2 dice together and count the total of the dots on the top faces

Question

2) If you roll 2 dice together and count the total of the dots on the top faces you get a distribution that has a mean = 7 and standard deviation = 2.45. Here is a histogram of that distribution:

Although this distribution is not Normal, the Central Limit Theorem tells us that if we draw samples of size = n from this population, (roll the dice n times, each time recording the sum of the top faces , and then find the average sum for each sample) the sample means will be Normally distributed.

a) Suppose we draw many samples of size n = 25 from this distribution. Those means will be distributed with a mean of _____________, and a standard deviation of ____________.

b) What proportion or percent of these means from samples of size n = 25 are between 6.5 and 7.5?

c) Suppose we change the sample to size n = 100 from the initial distribution. Those means will be distributed with a mean of _________, and a standard deviation of ____________.

d) How are the two standard deviations in a) and c) related? That is, an increase in the sample size from ______ to ______ results in a change in the standard deviation from _________ to _______________? What is the relationship/ratio?

e) What sample size would we need to use if we wanted to reduce the standard deviation for the distribution of the means of those samples to a value of .1?

Explanation / Answer

a) wioth a mean of 7 and std deviation of 2.45/(25)1/2 =0.49

b)P(6.5<X<7.5)=P((6.5-7)/.49<Z<(7.5-7)/0.49)=P(-1.0204<Z<1.0204)=0.8462-0.1538=0.6925

c)mean =7 ; std deviation =0.245

d)25 to 100 result in a change in the standard deviation from 0.49 to 0.245

e)here sample size =(2.45/0.1)2 =600.25~601

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