Two-sample t-tests Problem set A Worked solutions

In each case say which t-test to use, and state the assumptions if it is a two-sample test.

1. (a) Compare the mean reaction times of 14 boys and 16 girls, chosen at random.

   • The boys and the girls are two separate samples, so use a two-sample test.
   • Both populations are normally distributed. The two populations have equal variance.

2. (b) Test whether the mean mass of parcels differs from the stated 2 kg, using 20 parcels.

   • There is one sample and one stated value, so use a one-sample test. The population should be normal.

3. (c) Compare the mean fuel use of two models of car, tested on 10 cars and 12 cars.

   • Two separate samples of different sizes: a two-sample test.
   • Both populations are normally distributed. The two populations have equal variance.

State the null and alternative hypotheses for each test, defining μ1 and μ2.

1. (a) Does the mean time to solve a puzzle differ between adults (group 1) and children (group 2)?

   • Let μ1 and μ2 be the population mean times for adults and children.
   • H0: μ1 = μ2
   • H1: μ1 ≠ μ2 (different, so two-tailed)

2. (b) A vet thinks dogs on diet A (group 1) lose more weight on average than dogs on diet B (group 2).

   • Let μ1 and μ2 be the population mean weight loss on diet A and diet B.
   • H0: μ1 = μ2
   • H1: μ1 > μ2 (diet A loses more)

Two classes sit the same test. The marks are summarised. Test, at the 5% significance level, whether the mean mark of class 1 is greater than that of class 2. Assume the marks are normally distributed with equal variance. State the hypotheses, then find t and p and say whether H0 is rejected.

• H0: μ1 = μ2, H1: μ1 > μ2
• From the calculator, pooled: t = 2.15, p = 0.0201
• 0.0201 < 0.05, so reject H0.

The daily sales, in pounds, of a stall at two markets are summarised. Test, at the 5% significance level, whether the mean daily sales differ between the markets. Assume the sales are normally distributed with equal variance. State the hypotheses, then find t and p and say whether H0 is rejected.

• H0: μ1 = μ2, H1: μ1 ≠ μ2
• From the calculator, pooled: t = 0.908, p = 0.370
• 0.370 > 0.05, so do not reject H0.

The times, in minutes, that two types of paint take to dry are recorded on 6 samples of paint 1 and 7 samples of paint 2. Test, at the 5% significance level, whether the mean drying time of paint 1 is less than that of paint 2. Assume the times are normally distributed with equal variance. State the hypotheses, then find t and p and say whether H0 is rejected.

• H0: μ1 = μ2, H1: μ1 < μ2
• From the calculator, pooled: t = −2.57, p = 0.0131
• 0.0131 < 0.05, so reject H0.

A school teaches one group of students with a new method (group 1) and another group with the standard method (group 2). Their end-of-year marks are summarised. The school thinks the new method gives a higher mean mark. Test this belief at the 10% significance level.

1. (a) State two assumptions needed for the test.

   • Both populations are normally distributed. The two populations have equal variance.

2. (b) State the hypotheses.

   • Let μ1 and μ2 be the population mean mark for group 1 (new method) and group 2 (standard method).
   • H0: μ1 = μ2, H1: μ1 > μ2

3. (c) Find the t-statistic and the p-value.

   • From the calculator, pooled: t = 2.20, p = 0.0165

4. (d) State the conclusion in context.

   • 0.0165 < 0.1, so reject H0.
   • There is sufficient evidence, at the 10% significance level, that the mean mark is higher with the new method than with the standard method.

The weekly grocery spend, in pounds, of random samples of households in two towns is summarised. Test, at the 1% significance level, whether the mean weekly spend differs between the towns. Assume the spends are normally distributed with equal variance.

1. (a) State the hypotheses.

   • Let μ1 and μ2 be the population mean weekly grocery spend for group 1 (town X) and group 2 (town Y).
   • H0: μ1 = μ2, H1: μ1 ≠ μ2

2. (b) Find the t-statistic and the p-value.

   • From the calculator, pooled: t = 3.42, p = 0.00176

3. (c) State the conclusion in context.

   • 0.00176 < 0.01, so reject H0.
   • There is sufficient evidence, at the 1% significance level, that the mean weekly grocery spend differs between the two towns.

A test of whether the mean waiting time differs between two clinics gives a p-value of 0.027.

1. (a) State the conclusion at the 5% significance level, in context.

   • 0.0270 < 0.05, so reject H0.
   • There is sufficient evidence, at the 5% significance level, that the mean waiting time differs between the two clinics.

2. (b) State the conclusion at the 1% significance level, in context.

   • 0.0270 > 0.01, so do not reject H0.
   • There is insufficient evidence, at the 1% significance level, that the mean waiting time differs between the two clinics.

A typing teacher compares the number of words typed per minute by 8 students who took a typing course (group 1) and 7 students who did not (group 2). She believes the course increases the mean speed.

1. (a) State two assumptions needed.

   • Both populations are normally distributed. The two populations have equal variance.

2. (b) State the hypotheses.

   • Let μ1 and μ2 be the population mean typing speed for group 1 (course) and group 2 (no course).
   • H0: μ1 = μ2, H1: μ1 > μ2

3. (c) Test the claim at the 5% significance level. Give t, p and the conclusion in context.

   • From the calculator, pooled: t = 3.10, p = 0.00422
   • 0.00422 < 0.05, so reject H0.
   • There is sufficient evidence, at the 5% significance level, that the mean typing speed is higher after the course.

A researcher compares the memory scores of 13 students who studied with music (group 1) and 16 students who studied in silence (group 2). She thinks music improves the mean score. Test this at the 10% significance level, assuming normal populations with equal variance.

1. (a) State the hypotheses.

   • Let μ1 and μ2 be the population mean memory score for group 1 (music) and group 2 (silence).
   • H0: μ1 = μ2, H1: μ1 > μ2

2. (b) Find the t-statistic and the p-value.

   • From the calculator, pooled: t = 0.935, p = 0.179

3. (c) State the conclusion in context.

   • 0.179 > 0.1, so do not reject H0.
   • There is insufficient evidence, at the 10% significance level, that the mean memory score is higher with music than in silence.