Hypothesis testing (normal) Worked Solutions — A-Level Maths

Fully worked, step-by-step solutions to A-Level Hypothesis testing (normal) questions. See exactly how to solve problems on sampling-distribution, sample-mean, variance, standard-error.

sampling-distributionsample-meanvariancestandard-errorz-testtest-statistic
A-Level70 questionsStep-by-step solutions
Question 1
2 markseasy
The random variable X∼N(50, 36)X\sim N(50,\ 36). A random sample of 44 observations is taken. Write down the distribution of the sample mean Xˉ\bar{X}.

Worked solution

  1. Write down the population distribution

    X∼N(50, 36)X\sim N(50,\ 36)

    The individual observations are normally distributed.

  2. Apply the sample-mean result

    Xˉ∼N ⁣(μ, σ2n)\bar{X}\sim N\!\left(\mu,\ \dfrac{\sigma^2}{n}\right)

    The mean of a normal sample is normal with variance \(\sigma^2/n\).

  3. State the distribution of the sample mean

    Xˉ∼N ⁣(50, 9)\bar{X}\sim N\!\left(50,\ 9\right)

    Substituting \(\mu\), \(\sigma^2\) and \(n\) gives the distribution of \(\bar{X}\).

Answer
Xˉ∼N ⁣(50, 9)\bar{X}\sim N\!\left(50,\ 9\right)
Question 2
2 markseasy
The random variable X∼N(100, 100)X\sim N(100,\ 100). A random sample of 55 observations is taken. Write down the distribution of the sample mean Xˉ\bar{X}.

Worked solution

  1. Write down the population distribution

    X∼N(100, 100)X\sim N(100,\ 100)

    The individual observations are normally distributed.

  2. Apply the sample-mean result

    Xˉ∼N ⁣(μ, σ2n)\bar{X}\sim N\!\left(\mu,\ \dfrac{\sigma^2}{n}\right)

    The mean of a normal sample is normal with variance \(\sigma^2/n\).

  3. State the distribution of the sample mean

    Xˉ∼N ⁣(100, 20)\bar{X}\sim N\!\left(100,\ 20\right)

    Substituting \(\mu\), \(\sigma^2\) and \(n\) gives the distribution of \(\bar{X}\).

Answer
Xˉ∼N ⁣(100, 20)\bar{X}\sim N\!\left(100,\ 20\right)
Question 3
2 markseasy
The random variable X∼N(20, 16)X\sim N(20,\ 16). A random sample of 88 observations is taken. Write down the distribution of the sample mean Xˉ\bar{X}.

Worked solution

  1. Write down the population distribution

    X∼N(20, 16)X\sim N(20,\ 16)

    The individual observations are normally distributed.

  2. Apply the sample-mean result

    Xˉ∼N ⁣(μ, σ2n)\bar{X}\sim N\!\left(\mu,\ \dfrac{\sigma^2}{n}\right)

    The mean of a normal sample is normal with variance \(\sigma^2/n\).

  3. State the distribution of the sample mean

    Xˉ∼N ⁣(20, 2)\bar{X}\sim N\!\left(20,\ 2\right)

    Substituting \(\mu\), \(\sigma^2\) and \(n\) gives the distribution of \(\bar{X}\).

Answer
Xˉ∼N ⁣(20, 2)\bar{X}\sim N\!\left(20,\ 2\right)
Question 4
2 markseasy
The random variable X∼N(75, 144)X\sim N(75,\ 144). A random sample of 99 observations is taken. Write down the distribution of the sample mean Xˉ\bar{X}.

Worked solution

  1. Write down the population distribution

    X∼N(75, 144)X\sim N(75,\ 144)

    The individual observations are normally distributed.

  2. Apply the sample-mean result

    Xˉ∼N ⁣(μ, σ2n)\bar{X}\sim N\!\left(\mu,\ \dfrac{\sigma^2}{n}\right)

    The mean of a normal sample is normal with variance \(\sigma^2/n\).

  3. State the distribution of the sample mean

    Xˉ∼N ⁣(75, 16)\bar{X}\sim N\!\left(75,\ 16\right)

    Substituting \(\mu\), \(\sigma^2\) and \(n\) gives the distribution of \(\bar{X}\).

Answer
Xˉ∼N ⁣(75, 16)\bar{X}\sim N\!\left(75,\ 16\right)
Question 5
2 markseasy
The random variable X∼N(30, 25)X\sim N(30,\ 25). A random sample of 2525 observations is taken. Write down the distribution of the sample mean Xˉ\bar{X}.

Worked solution

  1. Write down the population distribution

    X∼N(30, 25)X\sim N(30,\ 25)

    The individual observations are normally distributed.

  2. Apply the sample-mean result

    Xˉ∼N ⁣(μ, σ2n)\bar{X}\sim N\!\left(\mu,\ \dfrac{\sigma^2}{n}\right)

    The mean of a normal sample is normal with variance \(\sigma^2/n\).

  3. State the distribution of the sample mean

    Xˉ∼N ⁣(30, 1)\bar{X}\sim N\!\left(30,\ 1\right)

    Substituting \(\mu\), \(\sigma^2\) and \(n\) gives the distribution of \(\bar{X}\).

Answer
Xˉ∼N ⁣(30, 1)\bar{X}\sim N\!\left(30,\ 1\right)

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