STA169 · TU past paper
Statistics I 2079 question paper
The complete TU 2079 exam paper for Statistics I (STA169), all 12 questions with solved model answers written to the mark scheme.
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- 110 marksNumericalMeasures of dispersionHideAnswer
Measures of Dispersion
Measures of dispersion quantify the spread or variability of data values about a central value. Central tendency tells us where data centers; dispersion tells us how scattered the data is. Small dispersion means values cluster tightly; l...
- 210 marksNumericalKarl Pearson's coefficient of correlationHideAnswer
Question
What are the assumptions of Pearson's correlation? Bradford Electric Illuminating is studying the relationship between kilowatt-hours (thousands) used and the number of rooms in private single-family residences. A random sample of 5 houses yields the following.
$$\begin{array}{c|ccccc} \text{Number of rooms} & 11 & 10 & 14 & 7 & 9 \ \hline \text{Kilowatts-hours (thousands) used} & 9 & 8 & 11 & 5 & 9 \ \end{array}$$
i. Find the correlation coefficient between the number of rooms and kilowatts-hours used. Interpret the result.
ii. Determine the regression equation of kilowatts hours used on the number of rooms. Interpret the value of the regression coefficient.
iii. Estimate the kilowatts-hours used for an 8 rooms house.
[10]
- Linear relationship: The two variables X and Y have a linear relationship. 2. Normality: The variables follow a bivariate normal distribution. 3. Continuous, quantitative data: Both variables are measured on an interval or ratio scale...
- 310 marksNumericalDiscrete distributionsHideAnswer
Question
Under what conditions does Binomial distribution tend to Poisson distribution? The number of telephone calls received during the month of May is summarized in the following table. Fit the Poisson distribution.
$$\begin{array}{c|ccccc} \text{Number of telephone calls per day} & 0 & 1 & 2 & 3 & 4 \ \hline \text{Number of days} & 8 & 12 & 18 & 13 & 9 \ \end{array}$$
[10]
Number of calls per day (x) 0 1 2 3 4 ------------------ Number of days (f) 8 12 18 13 9 $N = \sum f = 8 + 12 + 18 + 13 + 9 = 60$ --- The Binomial distribution $B(n, p)$ tends to the Poisson distribution when: 1. Number of trials is very...
- 45 marksNumericalMeasures of central tendencyHideAnswer
The following table shows the marks obtained by 130 students in computer science.i. Find the appropriate measure of central tendency. ii. Compute the minimum marks obtained by the top 20% of students. [5]
- Total number of students: $N = 130$ - Subject: Computer Science marks - Required: (i) appropriate measure of central tendency, (ii) minimum marks of top 20% of students Missing data: The actual frequency distribution table (class inter...
- 55 marksNumericalMeasures of dispersionHideAnswer
Question
Two batsmen A and B made the following runs in a series of cricket matches. Who is a more consistent player, and why?
$$\begin{array}{c|ccccc} A & 10 & 0 & 56 & 80 & 24 \ B & 36 & 37 & 45 & 28 & 29 \ \end{array}$$
[5]
- Batsman A runs: $10, 0, 56, 80, 24$ (n = 5) - Batsman B runs: $36, 37, 45, 28, 29$ (n = 5) Consistency is measured by the Coefficient of Variation (CV); the lower CV indicates the more consistent player. $$CV = \frac{\sigma}{\bar{x}} ...
- 65 marksNumericalConcepts of probabilityHideAnswer
Define conditional probability. A problem of statistics is given to three students, A, B, and C whose chances of solving the problem are in a ratio of 2:3:5. Find the probability that (i) none of them solve the problem (ii) the problem will be solved. [5]
Conditional probability is the probability of an event $A$ occurring given that another event $B$ has already occurred. It is defined as: $$P(AB) = \frac{P(A \cap B)}{P(B)}, \quad P(B) 0$$ --- - Ratio of chances of solving:
- 75 marksNumericalBayes theoremHideAnswer
A factory has three machines M1, M2, and M3 producing a large number of computer chips. Of the total daily production of items, 50% is produced on M1, 20% on M2, and 30% on M3. Records show that 4% of chips produced on M1 are defective, 2% produced of chips produced on M2 are defective and 2.5% of chips produced on M3 are defective. The occurrence of a defective chip is independent of all other chips. One chip is chosen at random from a day's total production. (i) Show that the probability of being defective is 0.0315 (ii) Given that it is defective, find the probability that was produced on machine M1. [5]
Machine Production Share Defective Rate ------------------------------------------ M1 $P(M1) = 0.50$ $P(DM1) = 0.04$ M2 $P(M2) = 0.20$ $P(DM2) = 0.02$ M3 $P(M3) = 0.30$ $P(DM3) = 0.025$ Let $D$ = event that a randomly chosen chip is defe...
- 85 marksNumericalMathematical expectation of a random variaHideAnswer
A coin is tossed two times and if X denotes the number of heads obtained, find (i) $E(x)$ (ii) $E(x^2)$ (iii) $V(x)$ (iv) $E(2X^2 + 3x - 5)$ [5]
- Fair coin tossed 2 times - X = number of heads - Possible values: X = 0, 1, 2 - Sample space: {TT, HT, TH, HH}, each equally likely with probability 1/4 X Outcomes P(X) ------------------- 0 TT 1/4 1 HT, TH 1/2 2 HH 1/4 Check:
- 95 marksNumericalDiscrete distributionsHideAnswer
The mean and variance of the number of flights arriving late in a day are 2 and 1.6 respectively. Assuming binomial distribution, are those values consistent? If yes, find the probability that (i) none of the flights are late today and (ii) at least one flight is late today. [5]
- Mean $= np = 2$ - Variance $= npq = 1.6$ For a binomial distribution, variance < mean always (since $q < 1$). Here mean $= 2$, variance $= 1.6$, and $1.6 < 2$, so the values are consistent. $$q = \frac{npq}{np} = \frac{1.6}{2} = 0.8$$ ...
- 105 marksNumericalContinuous distributionHideAnswer
What do you mean by normal distribution? From a batch of 10000, the lifetime of laptop batteries has a normal distribution with a mean of 40 months and a standard deviation of 8 months. What is the probability that a laptop selected at random will have life time (i) more than 50 months? (ii) between 40 and 50 months? [5]
Parameter Value ------------------ Batch size (N) 10,000 Mean (μ) 40 months Standard Deviation (σ) 8 months Required: (i) $P(X 50)$, (ii) $P(40 < X < 50)$ --- A normal distribution is a continuous probability distribution whose graph is ...
- 115 marksTypes of samplingHideAnswer
What do you understand by sampling? Differentiate between a simple random variable and stratified random sampling. [5]
Sampling is the process of selecting a subset of units (called a sample) from a larger group (called the population) in order to draw conclusions or make inferences about the entire population. Since studying every unit in a population i...
- 125 marksfive number summaryHideAnswer
Write notes on any two: i. Nominal and ordinal scale ii. Kurtosis iii. Five number summary [5]
--- Nominal scale is the simplest and lowest level of measurement scale. It is a system of assigning numbers or symbols to objects or events in order to distinguish one from another and label them. Key characteristics: - The symbols or n...