Sign in to jmaths to print this page or save it as a PDF: jmaths-8ce638778fd6.herokuapp.com/maths/ai-hl/statistics-2/ahl-8-8/problems/b/

jmaths

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 B

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

Each question matches the question with the same number in set A. Do the ones you got wrong there.

Difficulty: first skills applying them stretch

Skills

  1. A normal population has mean 85 and variance 63. A random sample of 7 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 \(X\sim N(64,\,9^2)\). A sample of 36 independent observations is taken. Find \(P(\bar X<62)\).

  3. A random variable X has mean 120 and variance 200. 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 = 60.
    2. (b) The distribution of X is not known and n = 25.
    3. (c) X is normally distributed and n = 16.

In context

  1. The number of pages in a student's essay is normally distributed with mean 8.5 and standard deviation 1.5. A random sample of 5 essays is chosen. Find the probability that the mean number of pages is less than 8.

  2. The bill at a cafe has a skewed distribution with mean \$14.20 and standard deviation \$6.50. A random sample of 80 bills is taken. Find the probability that the mean bill is more than \$15.

  3. The mass of a bag of gravel has mean 24.8 kg and standard deviation 1.6 kg, with an unknown distribution. A lorry carries 55 randomly chosen bags. Find the probability that their total mass is more than 1350 kg.

Stretch

  1. The daily number of visitors to a park is normally distributed with mean 200 and standard deviation 40.

    1. (a) Find the probability that one randomly chosen value is more than 215.
    2. (b) Find the probability that the mean of a random sample of 16 values is more than 215.
    3. (c) Explain why your two answers are so different.

  2. A lab measures the concentration of a solution in mg per litre. The readings are normally distributed with mean 500 and standard deviation 60. The lab reports the mean of several readings.

    1. (a) Find the probability that the sample mean is within 15 of 500 when the sample has 9 observations.
    2. (b) Find the same probability when the sample has 36 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