Testing a proportion Problem set B One question at a time

Write down suitable hypotheses for each test. Define p in words.

1. (a) A cinema says 35% of its tickets are sold online. The owner thinks fewer tickets are sold online now.

   • Let p be the probability that a ticket is sold online.
   • H0: p = 0.35, H1: p < 0.35 (the owner thinks the proportion has decreased)

2. (b) A farmer says 12% of her eggs are cracked on delivery. A shop thinks that more eggs are cracked.

   • Let p be the probability that an egg is cracked.
   • H0: p = 0.12, H1: p > 0.12 (the shop thinks the proportion is higher)

3. (c) A die is said to show a six with probability 1/6. A player thinks it shows a six less often.

   • Let p be the probability that the die shows a six.
   • H0: p = 1/6, H1: p < 1/6 (the player thinks a six is less likely)

For each test, X ~ B(n, p). Find the p-value and state the conclusion.

1. (a) H0: p = 0.55, H1: p < 0.55, n = 30, significance level 5%, observed value x = 12.

   • Under H0, X ~ B(30, 0.55).
   • p-value = P(X ≤ 12) = 0.0714
   • 0.0714 > 0.05, so do not reject H0. Insufficient evidence.

2. (b) H0: p = 0.1, H1: p > 0.1, n = 40, significance level 5%, observed value x = 8.

   • Under H0, X ~ B(40, 0.1).
   • p-value = P(X ≥ 8) = 1 − P(X ≤ 7) = 0.0419
   • 0.0419 < 0.05, so reject H0. Sufficient evidence that p is greater than 0.1.

Find the critical region for each test.

1. (a) X ~ B(50, p), H0: p = 0.6, H1: p < 0.6, significance level 5%.

   • Under H0, X ~ B(50, 0.6). The mean is 30.0.
   • From the calculator, cumulative probabilities near the 5% mark.
   • P(X ≤ 24) = 0.0573 > 0.05
   • P(X ≤ 23) = 0.0314 < 0.05
   • Critical region: X ≤ 23

2. (b) X ~ B(45, p), H0: p = 0.3, H1: p < 0.3, significance level 1%.

   • Under H0, X ~ B(45, 0.3). The mean is 13.5.
   • From the calculator, cumulative probabilities near the 1% mark.
   • P(X ≤ 7) = 0.0209 > 0.01
   • P(X ≤ 6) = 0.00797 < 0.01
   • Critical region: X ≤ 6

Find the critical region for each test.

1. (a) X ~ B(70, p), H0: p = 0.25, H1: p > 0.25, significance level 5%.

   • Under H0, X ~ B(70, 0.25). The mean is 17.5.
   • From the calculator, cumulative probabilities near the 5% mark.
   • P(X ≥ 24) = 0.0523 > 0.05
   • P(X ≥ 25) = 0.0299 < 0.05
   • Critical region: X ≥ 25

2. (b) X ~ B(40, p), H0: p = 0.7, H1: p > 0.7, significance level 10%.

   • Under H0, X ~ B(40, 0.7). The mean is 28.0.
   • From the calculator, cumulative probabilities near the 10% mark.
   • P(X ≥ 32) = 0.111 > 0.1
   • P(X ≥ 33) = 0.0553 < 0.1
   • Critical region: X ≥ 33

A school says that 30% of its pupils walk to school. After a new bus route starts, a teacher thinks the proportion who walk has decreased. In a random sample of 45 pupils, 8 walk. Test her belief at the 5% significance level.

1. (a) Define X and write down hypotheses.

   • Let X be the number of pupils who walk to school out of 45. X ~ B(45, p).
   • H0: p = 0.3, H1: p < 0.3

2. (b) Find the p-value.

   • Under H0, X ~ B(45, 0.3).
   • p-value = P(X ≤ 8) = 0.0471

3. (c) State the conclusion in context.

   • 0.0471 < 0.05, so reject H0.
   • There is sufficient evidence, at the 5% significance level, that the proportion of pupils who walk to school has decreased.

A bus company says that 80% of its buses on one route arrive on time. Commuters think the proportion is lower. They record 60 buses and count those that arrive on time. Use a 5% significance level.

1. (a) Define X and write down hypotheses.

   • Let X be the number of buses that arrive on time in a sample of 60. X ~ B(60, p).
   • H0: p = 0.8, H1: p < 0.8

2. (b) Find the critical region.

   • Under H0, X ~ B(60, 0.8).
   • P(X ≤ 43) = 0.0772 > 0.05
   • P(X ≤ 42) = 0.0427 < 0.05
   • Critical region: X ≤ 42

3. (c) The sample has 39 buses that arrive on time. State the conclusion.

   • 39 is in the critical region, so reject H0.
   • There is sufficient evidence, at the 5% significance level, that the proportion of buses that arrive on time is less than 0.8.

The graph shows the probabilities for X ~ B(30, 0.3). A test of H0: p = 0.3 against H1: p > 0.3 is carried out on a random sample of 30 at the 5% significance level.

1. (a) Find the critical region.

   • Under H0, X ~ B(30, 0.3).
   • P(X ≥ 13) = 0.0845 > 0.05
   • P(X ≥ 14) = 0.0401 < 0.05
   • Critical region: X ≥ 14

2. (b) Find the actual probability of rejecting H0 when H0 is true.

   • P(X ≥ 14) = 0.0401
   • The actual significance level is 4.01%.

3. (c) Explain why this is not equal to 5%.

   • Including 13 gives P(X ≥ 13) = 0.0845, which is over 5%.
   • A discrete distribution jumps from one probability to the next, so a critical region cannot have probability exactly 5%.
   • The probability of the region is below the significance level.

A multiple-choice quiz is tested for guessing: H0: p = 0.25, H1: p > 0.25, at the 10% significance level. For each number of questions n, find the critical region and its actual probability when H0 is true.

1. (a) n = 8

   • Under H0, X ~ B(8, 0.25).
   • P(X ≥ 4) = 0.114 > 0.1
   • P(X ≥ 5) = 0.0273 < 0.1
   • Critical region: X ≥ 5, with probability 0.0273

2. (b) n = 16

   • Under H0, X ~ B(16, 0.25).
   • P(X ≥ 6) = 0.190 > 0.1
   • P(X ≥ 7) = 0.0796 < 0.1
   • Critical region: X ≥ 7, with probability 0.0796

3. (c) n = 24

   • Under H0, X ~ B(24, 0.25).
   • P(X ≥ 9) = 0.121 > 0.1
   • P(X ≥ 10) = 0.0547 < 0.1
   • Critical region: X ≥ 10, with probability 0.0547