CAIE S2 (Statistics 2) 2020 Specimen

Leaves from a certain type of tree have lengths that are distributed with standard deviation 3 cm. A random sample of 6 of these leaves is taken and the mean length of this sample is found to be 8 cm.

1. Calculate a 95\% confidence interval for the population mean length. [3]
2. Write down the probability that the whole 95\% confidence interval will lie below the population mean. [1]

Describe briefly how to use a random number generator to obtain a sample of 10 students from a group of 50 students. [3]

The number of calls received at a small call centre has a Poisson distribution with mean 2 calls per 5 minute period.

1. Find the probability exactly 4 calls in a 10 minute period. [2]
2. Find the probability at least 3 calls in a 3 minute period. [3]

The number of calls received at a large call centre has a Poisson distribution with mean 4 calls per 5 minute period.

1. [(c)] Use an approximation to find the probability that the number of calls received in a 5 minute period is between 4 and 9 inclusive. [4]

The lifetimes, in hours, of light bulbs have an exponential distribution with parameter \(\frac{1}{500}\). Each bulb is tested and rejected if the lifetime is less than 500 hours.

1. Find the probability that a bulb of this type has a lifetime of more than 500 hours. [4]
2. Find the probability that the lifetime is at least three times the expected lifetime. [6]

The diagram shows the graph of the probability density function of a random variable \(X\), where $$f(x) = \begin{cases} \frac{1}{6}(3x - x^2) & 0 \leq x \leq 3, \\ 0 & \text{otherwise}. \end{cases}$$ \includegraphics{figure_1}

1. State the values of E(\(X\)) and Var(\(X\)). [4]
2. State the values of P(\(0.5 < X < 1\)). [1]
3. Given that P(\(1 < X < 2\)) = \(\frac{13}{27}\), find P(\(X > 2\)). [2]

At a certain hospital it was found that the probability that a patient did not arrive for an appointment was 0.1. The hospital's business model assumed that this probability will be reduced. They wish to test whether this probability is now less than 0.1. A random sample of 50 appointments is selected and the number of patients that did not arrive is noted. This figure is used as a test statistic at the 5\% significance level.

1. Explain why this test is a one-tailed test and state suitable null and alternative hypotheses. [2]
2. Use a binomial distribution to find the critical region and find the probability of a Type I error. [5]
3. In fact 3 patients out of the 50 did not arrive. State the conclusion of the test, explaining your answer. [2]

The mean weight of bags of carrots is \(\mu\) kilograms. An inspector wishes to test whether \(\mu = 20\). He weighs a random sample of 6 bags and the results are summarised as follows: $$\Sigma x = 430 \quad \Sigma x^2 = 40$$ Carry out the test at the 5\% significance level. [7]