Statistics and Probability Answers

From the frequency distribution table below calculate;
i. The harmonic mean
ii. The geometric mean
iii. The mode
Class 25-29 30-34 35-39 40-44 45-49 50-54
Frequency 3 9 13 10 7 2

The masses of packages from a particular machine are normally distributed with a mean of 200g and a standard deviation of 2g. Find the probability that a randomly selected package from the machine weighs
(i) Less than 196g
(iii) Between 198.5g and 199.5g

The table below shows discrete frequency distribution data. Use it to answer the questions that follow.
Class 0-4 5-9 10-14 15-19 20-24 25-29 30-34 35-39
Frequency 5 8 10 12 7 6 3 2

Compute;
(i) Mode of the distribution
(ii) The 7th decile
(iii)The third quartile

The exponential distribution with rate parameter μ > 0 is a continuous distribution on [0, ∞) with density

f(t) = μ exp(−μt), t > 0

1. Compute the cumulative distribution function defined by

F(t) := P(X ∈ [0,t]).

2. Compute P(s < X ≤ t).

3) Find P( X∈ [1,2] ∪[3,4] ).

4) Compute the conditional probability P (X ∈ [3, 4] | X ∈ [1, 4] )

5) Compute the conditional probability P (X > t + s | X > s ) for s, t ≥ 0.

6) Find the mean of the exponential distribution with rate parameter μ > 0.

A recent article in the Myrtle Beach Sun Times reported that the mean labor cost to repair a color television is $90.00 with a standard deviation of $22.00. Monte's TV sales and service completed repairs on two sets this morning. The labor cost for the first was $75 and it was $100 for the second. Compute the z values for each and comment on your findings.

Are there social class differences in the number of traffic violations? Here are the sample means and standard deviations for the number of tickets given for each group: lower class (mean = 3.2, sd = 1.4), middle class (mean = 10.2, sd = 2.3), and upper class (mean = 4.5, sd = 1.1). Assume you had 50 drivers in each group. If used ANOVA to conduct this analysis, what is the null hypothesis?

a) The following data relates to daily bill (in Kshs) on consumption of a certain commodity for 60 households

Daily bills (Kshs) No. of households
10 - 20 6
20 - 30 7
30 - 40 11
40 - 50 10
50 - 60 6
60 - 70 5
70 - 80 9
80 - 90 3
90 - 100 3


i) Calculate the mean
ii) Calculate the median
iii) Calculate the mode
iv) Calculate the standard deviation
v) Calculate the coefficient of skewness
vi) Comment on the skewness of this distribution
Calculate the coefficient of variation

The following data shows the different sizes of nails contained in a box bought by a
customer.


length(mm) Number of nails
10 to 14 10
15 to 19 12
20 to 24 18
25 to 29 16
30 to 34 4

(a) Find the followings:
i. Mean

ii. Median

iii. Standard deviation

(b) Calculate the Pearson’s coefficient of skewness and explain the distribution.

(c) Construct a cumulative frequency curve. From the cumulative frequency curve,
determine:

i. The first quartile

ii. The third quartile

iii. The percentage of nails with length exceeding 22mm.

A long straight road in a small town of New York is being used to monitor driving speeds.
The first 50 cars gave the results shown in Table 1 below. The figures are in km/h to
the nearest whole number.

45 60 71 57 67 78 90 81 78 91
65 88 67 66 50 68 67 73 63 94
48 90 79 95 65 70 75 61 54 67
57 59 85 84 70 81 40 59 76 59
77 78 97 75 72 77 54 68 67 74
Table 1

Answer the following questions:
(a) Find the range.

(b) Construct a frequency distribution for these data.

(c) Represent the data by a histogram and draw a frequency polygon on the
same graph. Hence, estimate the mode.

a) In one sample of observations, the sum of the squares of the observations of the sample values from sample mean was 120 and in the other sample of 12 observations it was 314. Test whether the difference is significant at 5% level of significance.