ZIMSEC A Level Statistics Paper 1 Past Papers

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Normal Distribution Probability Poisson Distribution Correlation and Regression Descriptive Statistics Continuous Random Variables Confidence Intervals Binomial and Geometric Distributionstime seriessummary statistics

Explained questions

A group of twenty people played a game. The frequency distribution of their scores is: score 1 with frequency 2, score 2 with frequency 5, score 4 with frequency 7, and score $x$ with frequency 6. If the mean score is 5, what is the value of $x$ ?
A group of twenty people played a game with scores 1, 2, 4 and 10, having frequencies 2, 5, 7 and 6 respectively (mean score 5). What is the variance of this distribution?
A head of school can contact parents by e-mail, letter or cellphone with probabilities 0.4, 0.1 and 0.5 respectively, using only one method. The probability that parents receive the message is 0.6 if e-mail is used, 0.8 if letter is used and 1 if cellphone is used. What is the probability that the parents receive the message?
A head of school contacts parents by e-mail, letter or cellphone with probabilities 0.4, 0.1 and 0.5 respectively. The probability parents receive the message is 0.6 for e-mail, 0.8 for letter and 1 for cellphone, so the overall probability of receiving the message is 0.82. Given that the parents received the message, what is the probability that it was sent by e-mail?
Which of the following is not one of the four traditional components of a time series?
Under which conditions can the Poisson distribution be used as an approximation to the Binomial distribution?
Potato seeds are packed in packets of 200 seeds, with on average 2% of the seeds in a packet rotten. A packet is substandard if it contains 5 or more rotten seeds. Using a Poisson approximation with mean $\lambda=200\times0.02=4$ , what is the probability that a packet is substandard, correct to 3 decimal places?
A random sample of 80 pockets of manure gives $\sum x=79.53$ and $\sum x^2=100.4621$ (in kg). Testing $H_0:\mu=1.10$ against $H_1:\mu<1.10$ , what is the value of the test statistic $t$ , to 2 decimal places?
A test of $H_0:\mu=1.10$ kg against $H_1:\mu<1.10$ kg for the mass of manure pockets gives a test statistic of $t=-1.82$ with 79 degrees of freedom, compared against a one-tailed 5% critical value of $-1.664$ . What is the correct conclusion?
Six sales representatives made the following numbers of calls to potential customers: 7, 6, 8, 6, 1, 2, with corresponding sales turnovers ($1000): 11, 10, 14, 12, 8, 9. What is the product moment correlation coefficient between number of calls and sales turnover, to 2 decimal places?
Six sales representatives made calls (7, 6, 8, 6, 1, 2) with corresponding sales turnovers ($1000): 11, 10, 14, 12, 8, 9. Using the least squares regression equation of turnover on number of calls, what is the gradient (slope) of the regression line, to 3 decimal places?
A student travels to school by bus, car, or on foot with probabilities $\frac{1}{6}$ , $\frac{1}{3}$ , and $\frac{1}{2}$ respectively. The probability of being late is $\frac{1}{5}$ if by bus, $\frac{1}{4}$ if by car, and $\frac{1}{20}$ if on foot. What is the probability that the student is early (not late) for school?
A student travels to school by bus, car, or on foot with probabilities $\frac{1}{6}$ , $\frac{1}{3}$ , and $\frac{1}{2}$ respectively. The probability of being late is $\frac{1}{5}$ if by bus, $\frac{1}{4}$ if by car, and $\frac{1}{20}$ if on foot. Given that the student arrives early on a particular day, what is the probability, as a fraction in lowest terms, that the student travelled on foot?
The times to the nearest minute taken by 12 students to complete a task are: 43, 45, 46, 42, 48, 42, 46, 55, 47, 42, 41, 44. What is the median time, in minutes?
The times to the nearest minute taken by 12 students to complete a task are: 43, 45, 46, 42, 48, 42, 46, 55, 47, 42, 41, 44. What is the interquartile range (Q3 minus Q1) of these times, in minutes?
The times to the nearest minute taken by 12 students to complete a task are 43, 45, 46, 42, 48, 42, 46, 55, 47, 42, 41, 44, giving Q1 = 42, median = 44.5, and Q3 = 46.5 minutes (IQR = 4.5). The upper fence for outliers is Q3 + 1.5×IQR = 53.25 minutes. What can be concluded about the value 55 and the overall shape of the distribution?
According to the Central Limit Theorem, for a sufficiently large sample size $n$ drawn from any population with mean $\mu$ and variance $\sigma^2$ , the distribution of the sample mean $\bar{X}$ is approximately
120 bags of mealie-meal have mean mass 750g and standard deviation 3.9g. What is the standard error of the mean, correct to 3 decimal places?
120 bags of mealie-meal have mean mass 750g and standard deviation 3.9g. Using a 95% confidence interval with $z = 1.96$ , what is the upper bound of the confidence interval for the mean mass, in grams, correct to 2 decimal places?
An ice cream vendor's sales are 40% chocolate, 35% vanilla and 25% peach. The probability of a cone sale is 0.80 for chocolate, 0.60 for vanilla and 0.40 for peach. What is the overall probability that a randomly selected sale is by cone?
The volume of dish washing liquid in a satchet is normally distributed with mean 400 ml and standard deviation 45 ml. The volume of fabric softener in a bottle is normally distributed with mean 650 ml and standard deviation 50 ml, independently of the satchets. Find the probability that the total volume of 4 randomly chosen satchets is less than the total volume of 2 randomly chosen bottles.
A machine packs washing powder into packets with mean weight 2 kg. The weights are normally distributed, and 10% of packets weigh less than 1.95 kg. Find the standard deviation of the packet weights, correct to 3 decimal places (in kg).
A machine packs washing powder into packets that are normally distributed with mean 2 kg and standard deviation 0.039 kg. Find the proportion of packets that weigh more than 2.10 kg.
A continuous random variable X has probability density function $f(x) = \dfrac{3}{8}x^2$ for $0 \leq x \leq 2$ . Find the value of the cumulative distribution function $F(1)$ .

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