Custom discrete distribution sample mean

Questions involving the sample mean of observations from a given discrete distribution (spinner, die, or other) with specified probabilities, where the CLT is applied using the given mean and variance.

5 questions · Standard +0.2

5.05a Sample mean distribution: central limit theorem

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CAIE S2 2020 June Q4

12 marks Standard +0.3

4 The score on one spin of a 5 -sided spinner is denoted by the random variable \(X\) with probability distribution as shown in the table.

\(x\)	0	1	2	3	4
\(\mathrm { P } ( X = x )\)	0.1	0.2	0.4	0.2	0.1

Show that \(\operatorname { Var } ( X ) = 1.2\).
The spinner is spun 200 times. The score on each spin is noted and the mean, \(\bar { X }\), of the 200 scores is found.
Given that \(\mathrm { P } ( \bar { X } > a ) = 0.1\), find the value of \(a\).
Explain whether it was necessary to use the Central Limit theorem in your answer to part (b).
Johann has another, similar, spinner. He suspects that it is biased so that the mean score is less than 2 . He spins his spinner 200 times and finds that the mean of the 200 scores is 1.86 . Given that the variance of the score on one spin of this spinner is also 1.2 , test Johann's suspicion at the 5\% significance level.

Edexcel FS1 2019 June Q3

6 marks Standard +0.8

A biased spinner can land on the numbers \(1,2,3,4\) or 5 with the following probabilities.

Number on spinner	1	2	3	4	5
Probability	0.3	0.1	0.2	0.1	0.3

The spinner will be spun 80 times and the mean of the numbers it lands on will be calculated. Find an estimate of the probability that this mean will be greater than 3.25
(6)

CAIE S2 2022 November Q5

6 marks Moderate -0.8

\(X\) is a random variable with distribution B(10, 0.2). A random sample of 160 values of \(X\) is taken.

Find the approximate distribution of the sample mean, including the values of the parameters. [3]
Hence find the probability that the sample mean is less than 1.8. [3]

CAIE S2 2011 June Q2

5 marks Standard +0.3

\(X\) is a random variable having the distribution \(\text{B}(12, \frac{1}{4})\). A random sample of 60 values of \(X\) is taken. Find the probability that the sample mean is less than 2.8. [5]

OCR S2 2012 January Q4

5 marks Standard +0.3

The discrete random variable \(H\) takes values 1, 2, 3 and 4. It is given that E(\(H\)) = 2.5 and Var(\(H\)) = 1.25. The mean of a random sample of 50 observations of \(H\) is denoted by \(\bar{H}\). Use a suitable approximation to find P(\(\bar{H} < 2.6\)). [5]