Answer ANY TWO (2) Questions from this section
Question 1
A. A survey was conducted to find the household size in Ghana during the
2010 Population Census. Find below the data of households by size.
Table 1: Household Size from Ghana Census of 2010
| Size of Household |
1 | 2 | 3 | 4 | 5 | 6 | 7 or more |
| Probability | 26.7% | 33.6% | 15.8% | 13.7% | 6.3% | 2.4% | 1.5% |
From the discrete probability distribution table given above, compute and
interpret the:
i. mean,
ii. variance
iii. standard deviation
[12 Marks]
B. Given that the number of calls coming per minute into a hotels
reservation center is Poison random variable with mean 3, find
i). The probability that no calls come in a given 1 minute period.
ii). Find the probability that at least two calls will arrive in a given
two minute period. [Assume that the number of calls arriving in
two different minutes is independent]. [8 Marks]
Question 2
A. Recently, research physicians have developed a knee-replacement
surgery process they believe will reduce the average patient recovery
time. The hospital board will not recommend the new procedure unless
there is substantial evidence that it is better than the existing procedure.
The current mean recovery rate for the standard procedure is 142 days,
with a standard deviation of 15 days. To test whether the new procedure
actually results in a lower mean recovery time, the procedure was
performed on a random sample of 36 patients and found a sample mean
of 140.2 days. Take the probability of making a Type I error as 5%.
i. Construct a hypotheses for this problem.
ii. Reach decision from the problem, given that the rejection region
specified by [latex]z[/latex][latex]_[latex]\alpha[/latex] = -1.645
iii. Draw a conclusion regarding the null hypothesis. [8 Marks]
B. Given that =2.5 and =0.866, Suppose x is a random variable best
described by a uniform probability that ranges from 1 to 4. Compute
P([latex]\mu[/latex] -[latex]\sigma[/latex][latex]\leq[/latex] x [latex]\geq[/latex][latex]\mu[latex] +[latex]\sigma[/latex]). [12 marks]
Question 3.
A. Road Safety Authority (RSA) concludes that the number of Broken-
Down Vehicles (BDVs) on road side in Ghana on each day of the week is
equal. A simple random sample of 700 BDVs from a recent year is
selected, and the results are shown below. At a significance level of 0.01,
is there enough evidence to support the RSA’s claim? [10 Marks]
Days |
Total number of BDVs Count | |
| Observe oi | Expected ei | |
| Sunday | 4,502 | 6,461.54 |
| Monday | 6,623 | 8,615.38 |
| Tuesday | 8,308 | 8,615.38 |
| Wednesday | 10,420 | 8,615.38 |
| Thursday | 11,032 | 8,615.38 |
| Friday | 10,754 | 8,615.38 |
| Saturday | 4,361 | 6,461.54 |
B(i). Convert the following data sets to stem and leaf. [6 Marks]
| 23 | 58 | 62 | 62 | 63 | 65 | 67 |
| 71 | 71 | 72 | 80 | 82 | 82 | 82 |
ii. Give two (2) examples each to distinguish the difference between
standard deviation and the range? [4 marks]
Question 4
A. A professor has conducted three different mid-term exams that are to be
graded on a 1,000-point scale. Before she uses the exams in a live class,
she wants to determine if the tests will yield the same mean scores. To
test this, a random sample of fourteen people is selected, categorized
according to their level of understanding so that each student takes each
test. From the results of the analysis given below: [12 marks]
| Source of variation | SS | Df | MS | F-Ratio | F-Critical |
| Between Blocks | 116,605.00 | 13 | X2 | X3 | 2.15 |
| Between samples | 241,912.70 | 2 | 120,956.40 | 12,279 | 3.4 |
| Within samples | X1 | 26 | 9,850.90 | ||
| Total | 614,641.61 | 41 |
i. Find the missing values X1, X2, and X3.
ii. State the hypothesis of the problem.
iii. Draw your conclusion.
B. In the end of first semester examinations, out of 20 students 5 scored
grade ‘A’. The management of UCC want to choose three students at
random out of the 20 students without replacement in order to award
them scholarship. Find the probability that all three are the ones who
scored an A’. [8 marks]
