Question

Use the Happy 1 variable for this exercise. Suppose someone claims the population mean is 55, and the standard deviation is 10.

PART 1 - For now, assume both of the claims about the population are correct.

1a. Given the assumed pop. mean and st.dev, calculate the probability of observing a value above the number for your first data point in the data set. (which is 36)

1b. Suppose you collected 8 new data points in a new sample. Calculate the probability that the mean of these 8 new data points is above the number for your first data point in your file.

1c. If this is a normally distributed variable, above what value should you find 70% of data points? How many of the values from your data set are above this value?

1d. If this is a normally distributed variable, between what two numbers (centered around the assumed mean) should you find 68% of data points? What percentage of your data points are between these numbers?

1e. Think about your answers to 1c and 1d. Does this variable appear to be normally distributed with this mean and standard deviation?

Happy1
36
18
66
43
28
39
47
40
24
46
48
57
36
58
39
62
43
65
74
36
39
44
61
50
47
63
60
38
45
51
55
46
68
32
42
38
61
45
31
32
44
30
29
62
49
54
64
38
49
55
28
53
55
52
50
54
76
28
49
70
29
34
77
40
50
40
56
54
36
51
42
71
45
53
55
37
51
36
39
36
51
40
51
52
53
33
66
37
76
67
55
46

Answer 1

1a) In this question we want to calculate P(X>36)

so

or

or

Hence the probability of observing a value greater than 36 = 0.97

1b) Now we have taken 8 new data points in a new sample. We will use the central limit theorem which defines the mean and standard deviation of a sample to be equal to and where

Mean of the population

Standard Deviation of the population

Sample Size

For this question as well, we want to calculate the probability

or,

or

Hence, if we take 8 new data points in anew sample then the probability that the sample mean will be greater than 36 will be approximately equal to 1.

1c) We want to know the value of a data point so that 70% of the data points are above that value.

In other words, we want to find a right-tailed confidence interval of the variable Happy1 with 70% confidence.

Hence, the Z-score of the value of the required data point should be more than 0.525 to have more 70% of data points lying above that value.

Let the required value of the data point be

So,

or

Hence, 70% of the data points will have a value greater than 60.25.

In the given data set only 17 data points of all has a value greater than 60.25 which is contradictory to the above statement of 70% of the datas being greater than 60.25.

1d) We again assume this dataset to be normally distributed. This time we want to know the range of values between which 68% of the data points will lie.

In other words we want to calculate a two tailed confidence interval for variable Happy1 with 68% confidence.

=   1 - 0.68 = 0.32

/2 = 0.16

Two tailed Value of z for = 0.32 = 0.995 (observed from the z-table)

Now, the expression for the 68% confidence interval for the variabe Happy1 can be given by

( - Z * , + Z * )

= ( 55 - 0.995 * 10 , 55 + 0.995 * 10)

= (45.05 , 64.95)

Hence, we can say that 68% of the values of the data set will lie between 45.05 and 64.95.

In the dataset, upon observation,we find that 40 value are between this interval. Which is less than 50% of the values and this answer is also contradictory with our question that 68% of the values should lie in between this interval.

1e) The answers in question c and d were highly contradictory to the assumption that the following dataset is normally distributed. Due to this wrong results, we have to say that the assumption is not true. That is, the given data set is not normally distributed.

We can show this by plotting these datas in excel.

As we can see the data points are not at all normally distributed.

Hence, the assumption is not true and that is why we are getting contradictory results in question c and d.

Thank You!!

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