Hello! I need help explaining the whole procedure, the meaning of the graphs and theorems (images) and how the final result was obtained
Thanks
Transcribed Image Text:FIGURE 7.4
Sampling distribution of
for n = 2 dice
FIGURE 7.5
MINITAB sampling distri-
bution of x for n = 3 dice
p(x-bar)
.15-
p(x-bar)
.10-
.05-
0
Using MINITAB, we generated the sampling distributions of when n = 3 and
n = 4. For n = 3, the sampling distribution in Figure 7.5 clearly shows the mound
shape of the normal probability distribution, still centered at μ = 3.5. Notice also that
the spread of the distribution is slowly decreasing as the sample size n increases.
Figure 7.6 dramatically shows that the distribution of is approximately normally
distributed based on a sample as small as n = 4. This phenomenon is the result of an
important statistical theorem called the Central Limit Theorem (CLT).
15-
.10-
.05-
0
Average of Two Dice
2
4
Average of Three Dice
5
x-bar
6
x-bar
Transcribed Image Text:1.4
FIGURE 7.3
Probability distribution for
x, the number appearing
on a single toss of a die
THE CENTRAL LIMIT THEOREM
The Central Limit Theorem states that, under rather general conditions, sums and
means of random samples of measurements drawn from a population tend to have an
approximately normal distribution. Suppose you toss a balanced die n = 1 time. The
random variable x is the number observed on the upper face. This familiar random
variable can take six values, each with probability 1/6, and its probability distribution
is shown in Figure 7.3. The shape of the distribution is flat or uniform and symmet-
ric about the mean = 3.5, with a standard deviation = 1.71. (See Section 4.8 and
Exercise 4.84.)
p(x)
1/6-
0
2
x
Now, take a sample of size n = 2 from this population; that is, toss two dice and
record the sum of the numbers on the two upper faces, Ex; = x₁ + x₂. Table 7.5 shows
the 36 possible outcomes, each with probability 1/36. The sums are tabulated, and
each of the possible sums is divided by n = 2 to obtain an average. The result is the
sampling distribution of x = Ex/n, shown in Figure 7.4. You should notice the dra-
matic difference in the shape of the sampling distribution. It is now roughly mound-
shaped but still symmetric about the mean μ = 3.5.
Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...
Expert Solution
Step 1
If n = 1 a single dice
If a single dice is thrown the probability of getting {1,2,3,4,5,6} = 1/6
Each value is having equal chance of occurrence then the probability is 1/6
Read through expert solutions to related follow-up questions below.
Follow-up Question
Hi! from the previous explanation I have doubts about how the probabilities of the table are found when using 1,2 and 3 dice. Why is p(x) decreasing as x increases?
by Bartleby Expert