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# Descriptive Statistics, Probability & Confidence Interval

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 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