Further hypothesis tests Worked Solutions — Further Maths Maths

Fully worked, step-by-step solutions to Further Maths Further hypothesis tests questions. See exactly how to solve problems on choosing-a-test, paired-data, paired-t-test, pooled-variance.

choosing-a-testpaired-datapaired-t-testpooled-variancetwo-sample-t-testf-test
Further Maths70 questionsStep-by-step solutions
Question 1
2 markseasy
Twelve members of a slimming club are each weighed at the start of a six-week course and again at the end of it: the same twelve people are weighed on both occasions. The club wishes to test whether the course changes the mean mass. Each observation in one sample is matched with exactly one observation in the other sample. The claim concerns the means of the two populations. The population variances are unknown. Both populations are assumed to be normal. Which of the following tests is appropriate?

Worked solution

  1. Decide whether the population variances are known

    the population variances are unknown\text{the population variances are unknown}

    A known population variance allows a zz-test; an unknown one forces a tt-test.

  2. Match the situation to the standard tests

    a matched-pairs design with an unknown variance\text{a matched-pairs design with an unknown variance}

    Each of the standard two-sample tests answers exactly one kind of question.

  3. State the appropriate test and its test statistic

    d=x−y,t=dˉ−μdsd/n,ν=n−1d=x-y,\quad t=\frac{\bar{d}-\mu_d}{s_d/\sqrt{n}},\quad \nu=n-1

    Each 'before' reading is tied to exactly one 'after' reading on the same person, so the two samples are not independent.

Answer
A paired tt-test on the differences d=x−yd=x-y, with ν=n−1\nu=n-1 degrees of freedom
Question 2
2 markseasy
Two independent random samples, of 99 tomatoes grown in greenhouse AA and 1111 tomatoes grown in greenhouse BB, are weighed; no tomato appears in both samples. The grower wishes to test whether the two greenhouses produce tomatoes of the same mean mass. No observation in one sample is matched with any particular observation in the other sample. The claim concerns the means of the two populations. The population variances are unknown. Both populations are normal and may be assumed to have a common variance, which must therefore be estimated from the two samples. Which of the following tests is appropriate?

Worked solution

  1. Decide whether the observations are paired

    the two samples are independent and of different sizes, so the data cannot be paired\text{the two samples are independent and of different sizes, so the data cannot be paired}

    Two samples are paired only if each observation in one is matched with exactly one in the other on the same unit.

  2. Decide whether the population variances are known

    the common population variance is unknown and must be estimated by pooling\text{the common population variance is unknown and must be estimated by pooling}

    A known population variance allows a zz-test; an unknown one forces a tt-test.

  3. Match the situation to the standard tests

    two independent samples, unknown but equal variances\text{two independent samples, unknown but equal variances}

    Each of the standard two-sample tests answers exactly one kind of question.

  4. State the appropriate test and its test statistic

    t=(xˉ−yˉ)−(μ1−μ2)sp1n1+1n2,ν=n1+n2−2t=\frac{(\bar{x}-\bar{y})-(\mu_1-\mu_2)}{s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}},\quad \nu=n_1+n_2-2

    With independent samples and an unknown common variance, the pooled estimate sp2s_p^2 supplies the standard error.

Answer
A two-sample (unpaired) tt-test for the difference of the means, using the pooled variance estimate and ν=n1+n2−2\nu=n_1+n_2-2 degrees of freedom
Question 3
2 markseasy
A laboratory buys a new pipette and wishes to know whether it dispenses liquid more consistently than the old one. Independent random samples of dispensed volumes are taken from each pipette and the two unbiased sample variances are computed. No observation in one sample is matched with any particular observation in the other sample. The claim concerns the variances of the two populations, not their means. The population variances are unknown. Both populations are assumed to be normal. Which of the following tests is appropriate?

Worked solution

  1. Decide whether the population variances are known

    both population variances are unknown and are estimated from the samples\text{both population variances are unknown and are estimated from the samples}

    A known population variance allows a zz-test; an unknown one forces a tt-test.

  2. Match the situation to the standard tests

    a claim about spread, not about location\text{a claim about spread, not about location}

    Each of the standard two-sample tests answers exactly one kind of question.

  3. State the appropriate test and its test statistic

    F=larger s2smaller s2,ν=(n1−1, n2−1)F=\frac{\text{larger }s^2}{\text{smaller }s^2},\quad \nu=(n_1-1,\ n_2-1)

    A claim about consistency is a claim about variance, and the ratio of two unbiased variance estimates has an FF distribution.

Answer
An FF-test for the equality of the two population variances, with (n1−1, n2−1)(n_1-1,\,n_2-1) degrees of freedom
Question 4
2 markseasy
Two filling machines have been in use for many years, and from long records the variances of the volumes they dispense are σ12=4\sigma_1^2=4 and σ22=9\sigma_2^2=9. Independent random samples of 5050 and 6060 bottles are taken and the mean volume of each sample is computed. The engineer wishes to test whether the two machines dispense the same mean volume. No observation in one sample is matched with any particular observation in the other sample. The claim concerns the means of the two populations. The population variances are known. Which of the following tests is appropriate?

Worked solution

  1. Decide whether the observations are paired

    the two samples are independent random samples of different sizes\text{the two samples are independent random samples of different sizes}

    Two samples are paired only if each observation in one is matched with exactly one in the other on the same unit.

  2. Decide whether the population variances are known

    both population variances are KNOWN, so nothing has to be estimated\text{both population variances are KNOWN, so nothing has to be estimated}

    A known population variance allows a zz-test; an unknown one forces a tt-test.

  3. Match the situation to the standard tests

    two independent samples with known variances\text{two independent samples with known variances}

    Each of the standard two-sample tests answers exactly one kind of question.

  4. State the appropriate test and its test statistic

    z=(xˉ1−xˉ2)−(μ1−μ2)σ12n1+σ22n2∼N(0,1)z=\frac{(\bar{x}_1-\bar{x}_2)-(\mu_1-\mu_2)}{\sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}}\sim N(0,1)

    When σ12\sigma_1^2 and σ22\sigma_2^2 are known there is no estimation, so the statistic is exactly standard normal and no tt distribution is needed.

Answer
A two-sample zz-test for the difference of the means, using the known population variances
Question 5
2 markseasy
Ten patients each have their cholesterol level measured before and after taking a drug for one month, giving ten 'before' readings xx and ten 'after' readings yy, one of each per patient. Each observation in one sample is matched with exactly one observation in the other sample. The claim concerns the means of the two populations. The population variances are unknown. Both populations are assumed to be normal. Which of the following statements about the correct test for these data is true?

Worked solution

  1. Decide what the two samples measure

    the same patients are measured twice, so the two samples are NOT independent\text{the same patients are measured twice, so the two samples are NOT independent}

    Paired data arise when the same unit is measured twice, or when units are matched deliberately.

  2. Decide which sample of numbers the test is carried out on

    di=xi−yi,i=1,2,…,10d_i=x_i-y_i,\quad i=1,2,\dots,10

    A paired test reduces the data to ONE sample of differences; an unpaired test keeps two samples.

  3. State the correct test and its degrees of freedom

    ν=n−1=9\nu=n-1=9

    Because the same patients appear in both samples, the between-patient variation cancels in the differences, and ν=n−1=9\nu=n-1=9.

Answer
The data are paired, so the test must be carried out on the nn differences d=x−yd=x-y, using t=dˉ−μdsd/nt=\dfrac{\bar{d}-\mu_d}{s_d/\sqrt{n}} with ν=n−1\nu=n-1

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