Two-sample t-tests Problem set B 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 heights of plants given fertiliser X (9 plants) and fertiliser Y (11 plants).

   • The two sets of plants are 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 waiting time at a clinic differs from the stated 15 minutes, using 25 patients.

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

3. (c) Compare the mean scores of two classes, with 22 and 19 students.

   • 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 daily step count differ between office workers (group 1) and teachers (group 2)?

   • Let μ1 and μ2 be the population mean daily step counts of office workers and teachers.
   • H0: μ1 = μ2
   • H1: μ1 ≠ μ2 (different, so two-tailed)

2. (b) A baker thinks loaves made with flour A (group 1) have a lower mean mass than loaves made with flour B (group 2).

   • Let μ1 and μ2 be the population mean mass of loaves made with flour A and flour B.
   • H0: μ1 = μ2
   • H1: μ1 < μ2 (flour A gives a lower mean)

Two brands of battery are timed while charging. The charging times, in hours, are summarised. Test, at the 5% significance level, whether the mean charging time of brand 1 is less than that of brand 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 = −2.59, p = 0.00790
• 0.00790 < 0.05, so reject H0.

The hours of sleep per night of random samples of students at two schools are summarised. Test, at the 5% significance level, whether the mean hours of sleep differ between the schools. Assume the hours 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 = −0.680, p = 0.500
• 0.500 > 0.05, so do not reject H0.

The heights, in cm, of 7 seedlings grown under light 1 and 6 seedlings grown under light 2 are recorded. Test, at the 5% significance level, whether the mean height under light 1 is greater than under light 2. Assume the heights 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 = 2.28, p = 0.0219
• 0.0219 < 0.05, so reject H0.

A coach gives one group of athletes training plan A (group 1) and another group plan B (group 2). The numbers of press-ups each athlete can do in one minute are summarised. The coach thinks plan A gives a higher mean. 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 number of press-ups for group 1 (plan A) and group 2 (plan B).
   • H0: μ1 = μ2, H1: μ1 > μ2

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

   • From the calculator, pooled: t = 2.43, p = 0.0103

4. (d) State the conclusion in context.

   • 0.0103 < 0.1, so reject H0.
   • There is sufficient evidence, at the 10% significance level, that the mean number of press-ups is greater after plan A than after plan B.

The daily screen time, in hours, of random samples of students in Year 10 and Year 12 is summarised. Test, at the 1% significance level, whether the mean daily screen time differs between the year groups. Assume the times are normally distributed with equal variance.

1. (a) State the hypotheses.

   • Let μ1 and μ2 be the population mean daily screen time for group 1 (Year 10) and group 2 (Year 12).
   • H0: μ1 = μ2, H1: μ1 ≠ μ2

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

   • From the calculator, pooled: t = 2.96, p = 0.00481

3. (c) State the conclusion in context.

   • 0.00481 < 0.01, so reject H0.
   • There is sufficient evidence, at the 1% significance level, that the mean daily screen time differs between Year 10 and Year 12.

A test of whether the mean life of two models of phone battery differs gives a p-value of 0.043.

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

   • 0.0430 < 0.05, so reject H0.
   • There is sufficient evidence, at the 5% significance level, that the mean battery life differs between the two models.

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

   • 0.0430 > 0.01, so do not reject H0.
   • There is insufficient evidence, at the 1% significance level, that the mean battery life differs between the two models.

An angler compares the masses, in kg, of 7 fish caught with bait A (group 1) and 8 fish caught with bait B (group 2). She believes bait A catches heavier fish on average.

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 mass of a fish for group 1 (bait A) and group 2 (bait B).
   • 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 = 3.02, p = 0.00490
   • 0.00490 < 0.05, so reject H0.
   • There is sufficient evidence, at the 5% significance level, that the mean mass of fish is greater with bait A.

A cycling club compares the distance, in km, ridden before a puncture with a new tyre (group 1, 14 tyres) and a standard tyre (group 2, 17 tyres). The club thinks the new tyre lasts longer. 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 distance before a puncture for group 1 (new tyre) and group 2 (standard tyre).
   • H0: μ1 = μ2, H1: μ1 > μ2

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

   • From the calculator, pooled: t = 1.04, p = 0.154

3. (c) State the conclusion in context.

   • 0.154 > 0.1, so do not reject H0.
   • There is insufficient evidence, at the 10% significance level, that the mean distance before a puncture is greater with the new tyre.