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# Descriptive Statistics, Probability & Confidence Interval

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1. Calculate the mean, median and standard deviation of the following data:
10, 16, 12, 20, 18, 209, 16, 11, 13, 14, 17, 20, 25

2. Prepare a relative frequency histogram of the following data, use 4 classes.
43, 35, 41, 41, 39, 38, 41, 40, 45, 43, 52, 36, 43, 38, 39, 44, 45, 30, 37, 40, 39, 41, 44, 46, 39, 40, 44, 37, 47, 48, 49, 51, 53, 52, 51, 39.

3. Calculate the probability of the following binomial distributions.
(a). P (X = 1) if n = 11 and p = 0.26
(b). P (X > 2) if n = 12, p = 0.42

4. For a Poisson distribution with mean 3, calculate the following probabilities.
(a). P {X < 4}
(b). P {X = 5}

5. Calculate the expected value and the variance of the random variable X having the following probabilities
X 20 25 30 40 50 60
P (X) 0.1 0.25 0.1 0.25 0.16 0.14

6. Suppose that a random variable X is distributed as normal with mean 100 and variance 9. Calculate the following probabilities.
(a). P (X > 105) (b). P (99 < X < 103)

7. Calculate 95% confidence interval for µ if = 25, s2 = 6 and n = 24. Assume the observations are from a normal population.

See attached file for problems.

##### Solution Summary

The solution provides step by step method for the calculation of descriptive statistics, probability and confidence interval for mean. Formula for the calculation and Interpretations of the results are also included. Interactive excel sheet is included. The user can edit the inputs and obtain the complete results for a new set of data.

##### Solution Preview

1. Calculate the mean, median and standard deviation of the following data:
10, 16, 12, 20, 18, 209, 16, 11, 13, 14, 17, 20, 25
X X^2
10 100
16 256
12 144
20 400
18 324
209 43681
16 256
11 121
13 169
14 196
17 289
20 400
25 625
401 46961

Mean, = 401/13
= 30.84615385
Median = item when the items are arranged in ascending or descending order of magnitude.
The observations can be arranged in ascending order as follows:
10, 11, 12, 13, 14, 16, 16, 17, 18, 20, 20, 25, 209
Therefore, Median = item = 7th item
= 16
Standard deviation = =
= 51.58390649
2. Prepare a relative frequency histogram of the following data, use 4 classes.
43, 35, 41, 41, 39, 38, 41, 40, 45, 43, 52, 36, 43, 38, 39, 44, 45, 30, 37, 40, 39, 41, 44, 46, 39, 40, 44, 37, 47, 48, 49, 51, 53, 52, 51, 39.

Data cumulative
lower upper midpoint width frequency percent frequency percent
30 < 36 33 6 2 5.6 2 5.6
36 < 42 39 6 17 47.2 19 52.8
42 < 48 45 6 10 27.8 29 80.6
48 < 54 51 6 7 19.4 36 100.0

36 100.0

3. Calculate the probability of the following binomial distributions.
(a). P (X = 1) if n = 11 and p = 0.26
(b). P (X > 2) if n = 12, p = 0.42
(a). Given X is binomial with n = 11 and p =0.26. The probability mass function binomial variable is given by . The probability for different values of x is given ...

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