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IB Mathematics: Applications and Interpretation HL · AHL 8: Statistics 2 · AHL 8.8

Name Date

The sample mean and the central limit theorem Problem set A

Write the distribution of the sample mean or the total, then use the calculator for the probability.

Difficulty: first skills applying them stretch

Skills

  1. A normal population has mean 52 and variance 40. A random sample of 8 observations is taken. \(\bar X\) is the sample mean.

    1. (a) State the distribution of \(\bar X\).
    2. (b) Write down the standard error.

  2. A random variable \(X\sim N(150,\,20^2)\). A sample of 16 independent observations is taken. Find \(P(\bar X<145)\).

  3. A random variable X has mean 28 and variance 49. Samples of independent observations of X are taken. Give the distribution of \(\bar X\) in each case, or say that it cannot be found.

    1. (a) The distribution of X is not known and n = 45.
    2. (b) The distribution of X is not known and n = 20.
    3. (c) X is normally distributed and n = 20.

In context

  1. The time for a kettle to boil is normally distributed with mean 3.4 minutes and standard deviation 0.5 minutes. A random sample of 4 boils is timed. Find the probability that the mean time is greater than 3.7 minutes.

  2. The waiting time at a clinic has a skewed distribution with mean 18 minutes and standard deviation 11 minutes. A random sample of 60 patients is taken. Find the probability that the mean waiting time is less than 15 minutes.

  3. The mass of a parcel has mean 3.6 kg and standard deviation 1.8 kg. A van is loaded with 45 randomly chosen parcels. Find the probability that their total mass is more than 170 kg.

Stretch

  1. The score on a reaction test is normally distributed with mean 60 and standard deviation 15.

    1. (a) Find the probability that one randomly chosen value is more than 66.
    2. (b) Find the probability that the mean of a random sample of 25 values is more than 66.
    3. (c) Explain why your two answers are so different.

  2. A scale gives readings that are normally distributed with mean 100 g and standard deviation 20 g. A technician takes several readings and works with their mean.

    1. (a) Find the probability that the sample mean is within 2 of 100 when the sample has 25 observations.
    2. (b) Find the same probability when the sample has 100 observations.
    3. (c) What happens to the probability as the sample size grows?

jmaths.xyz · AI HL · AHL 8.8 The sample mean and the central limit theorem