The problem i have is the most is the excersie questions! i have very bad englis
ID: 3135393 • Letter: T
Question
The problem i have is the most is the excersie questions! i have very bad english and cant understand. I can understand numbers though X)
Let X be a random variable representing the roll of a fair 6-sided die. Complete the following table which will represent the theoretical distribution of X (Value of the Die and its corresponding probability). This represents the distribution of your population, since we are examining the theoretical outcome of all possible rolls of the die.
Table 1
Die Value (x)
Probability that X = x
1
2
3
4
5
6
Now we will conduct our experiment:
Roll your die 4 times and calculate the average of these 4 rolls () in the first box (Roll #1) in the table provided below.
Now repeat the previous step 40 times. You should have 40 written in Table 2 . This is your sampling distribution.
Table 2
Roll #
1
2
3
4
5
6
7
8
9
10
Roll #
11
12
13
14
15
16
17
18
19
20
Roll #
21
22
23
24
25
26
27
28
29
30
Roll #
31
32
33
34
35
36
37
38
39
40
Now sort your 40 values of and complete Table 3 given below.
Table 3
Interval
Number of Rolls
Proportion out of 40 (Probability)
1 but less than 2
2 but less than 3
3 but less than 4
4 but less than 5
5 to 6
We will now construct two histograms on the top and bottom of this page: The top histogram will be our population from Table 1 (single roll of a die). The bottom histogram will be our sampling distribution of the sample mean from Table 3 (mean of n=4 rolls of a die). Remember, the horizontal axis represents the possible values of the random variable while the vertical axis represents the probability of getting the particular value.
Exercise Questions
Recall the premise of the Central Limit Theorem: The mean of a random sample will approximately follow a normal distribution with mean µ and standard error , regardless of the distribution of the population. The theory requires a sample size of at least 25 if the population distribution is unknown. However, because we know the distribution of the die and this distribution happens to be symmetric too, we can get away with a much smaller sample size (n=4) and still see how the Central Limit Theorem works.
We will now compare the theoretical results of rolling one die versus the experiment you performed: the mean of 4 rolls of a die.
Comment on the difference in appearance (bell curve, symmetric or skewed) between the top and bottom histograms. Does the Central Limit Theorem appear to be working here? Explain your answer.
In Minitab, type in the theoretical outcome for rolling a single die 300 times
(1,2,3,4,5,6,1,2,3,4,5,6,….1,2,3,4,5,6) in column C1. This supports our distribution that we created in Table 1 since each number has probability 50/300 = 1/6. Type in 1,2,3,4,5,6, then cut and paste this sequence so you have 300 values. This will be the population, or the theoretical results of rolling a single die many times.
Now type in the results of your experiment (the 40 means you obtained from Table 2) in column C2. Calculate the mean and standard deviations for both column C1 and C2 using Minitab Stats>Basic Stats>Display Descriptive Statistics. The first mean and standard deviation (C1, from the 300 rolls) will be µ (mu) and (sigma) which will be used in the Central Limit Theorem. The second mean and standard deviation (C2) will be your sample results. Compare the mean and standard deviations for both the population (C1) and your sample (C2) Are the means somewhat similar or different? Are the standard deviations somewhat similar or different?
Looking at the top and bottom histograms visually, which one appears to have less variability: the population (top graph) or the mean of the experiment (rolling the die n=4 times)? Does this visual result support your standard deviation comparison you got in problem #2? Explain why or why not.
Now we will verify if the Central Limit Theorem is working with regards to the standard deviation. Calculate / squareroot(n ), using the result from problem #2 for ( is the standard deviation for C1 and n is of course 4 since that was the sample size used to calculate the means for Table 2), to what your actual standard error of the mean is (the standard deviation of Column C2 that you found in problem
#2). How close are the theoretical results to what you actually obtained? Remember, nothing is perfect. Based on this comparison, do you believe the Central Limit Theorem is working here with regards to what was anticipated versus the actual results? Explain.
Die Value (x)
Probability that X = x
1
2
3
4
5
6
Explanation / Answer
Let X be a random variable representing the roll of a fair 6-sided die. Complete the following table which will represent the theoretical distribution of X (Value of the Die and its corresponding probability). This represents the distribution of your population, since we are examining the theoretical outcome of all possible rolls of the die.
Die has 6 outcomes and each outcome has probability 1/6.
Die value (x) probability 1 1|6 2 1|6 3 1|6 4 1|6 5 1|6 6 1|6 total 1Related Questions
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