Testing a proportion Problem set A One question at a time

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

1. (a) A bakery says 25% of its customers buy a cake. The manager thinks more customers buy a cake since a new display was added.

   • Let p be the probability that a customer buys a cake.
   • H0: p = 0.25, H1: p > 0.25 (the manager thinks the proportion has increased)

2. (b) A phone maker says 4% of its phones are returned. A reviewer thinks the proportion of returns is lower than this.

   • Let p be the probability that a phone is returned.
   • H0: p = 0.04, H1: p < 0.04 (the reviewer thinks the proportion is lower)

3. (c) A spinner is said to land on red with probability 1/5. A player thinks it lands on red more often.

   • Let p be the probability that the spinner lands on red.
   • H0: p = 1/5, H1: p > 1/5 (the player thinks red is more likely)

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

1. (a) H0: p = 0.4, H1: p < 0.4, n = 25, significance level 5%, observed value x = 5.

   • Under H0, X ~ B(25, 0.4).
   • p-value = P(X ≤ 5) = 0.0294
   • 0.0294 < 0.05, so reject H0. Sufficient evidence that p is less than 0.4.

2. (b) H0: p = 0.15, H1: p > 0.15, n = 50, significance level 10%, observed value x = 12.

   • Under H0, X ~ B(50, 0.15).
   • p-value = P(X ≥ 12) = 1 − P(X ≤ 11) = 0.0628
   • 0.0628 < 0.1, so reject H0. Sufficient evidence that p is greater than 0.15.

Find the critical region for each test.

1. (a) X ~ B(40, p), H0: p = 0.35, H1: p < 0.35, significance level 5%.

   • Under H0, X ~ B(40, 0.35). The mean is 14.0.
   • From the calculator, cumulative probabilities near the 5% mark.
   • P(X ≤ 9) = 0.0644 > 0.05
   • P(X ≤ 8) = 0.0303 < 0.05
   • Critical region: X ≤ 8

2. (b) X ~ B(60, p), H0: p = 0.5, H1: p < 0.5, significance level 1%.

   • Under H0, X ~ B(60, 0.5). The mean is 30.0.
   • From the calculator, cumulative probabilities near the 1% mark.
   • P(X ≤ 21) = 0.0137 > 0.01
   • P(X ≤ 20) = 0.00674 < 0.01
   • Critical region: X ≤ 20

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 ≥ 36) = 0.0540 > 0.05
   • P(X ≥ 37) = 0.0280 < 0.05
   • Critical region: X ≥ 37

2. (b) X ~ B(80, p), H0: p = 0.2, H1: p > 0.2, significance level 10%.

   • Under H0, X ~ B(80, 0.2). The mean is 16.0.
   • From the calculator, cumulative probabilities near the 10% mark.
   • P(X ≥ 21) = 0.107 > 0.1
   • P(X ≥ 22) = 0.0660 < 0.1
   • Critical region: X ≥ 22

A shop owner says that 20% of customers pay by card. A bank claims that the proportion has increased since it fitted a faster card reader. In a random sample of 60 customers, 17 pay by card. Test the bank's claim at the 5% significance level.

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

   • Let X be the number of customers who pay by card out of 60. X ~ B(60, p).
   • H0: p = 0.2, H1: p > 0.2

2. (b) Find the p-value.

   • Under H0, X ~ B(60, 0.2).
   • p-value = P(X ≥ 17) = 1 − P(X ≤ 16) = 0.0772

3. (c) State the conclusion in context.

   • 0.0772 > 0.05, so do not reject H0.
   • There is insufficient evidence, at the 5% significance level, that the proportion of customers who pay by card has increased.

A nursery says that 45% of its tulip bulbs produce a flower in the first year. After a cold winter, a gardener thinks the proportion is lower. She plants 50 bulbs and counts those that flower. Use a 5% significance level.

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

   • Let X be the number of plants with a flower in a sample of 50. X ~ B(50, p).
   • H0: p = 0.45, H1: p < 0.45

2. (b) Find the critical region.

   • Under H0, X ~ B(50, 0.45).
   • P(X ≤ 17) = 0.0765 > 0.05
   • P(X ≤ 16) = 0.0427 < 0.05
   • Critical region: X ≤ 16

3. (c) The sample has 17 plants with a flower. State the conclusion.

   • 17 is not in the critical region, so do not reject H0.
   • There is insufficient evidence, at the 5% significance level, that the proportion of plants with a flower is less than 0.45.

The graph shows the probabilities for X ~ B(25, 0.4). A test of H0: p = 0.4 against H1: p > 0.4 is carried out on a random sample of 25 at the 5% significance level.

1. (a) Find the critical region.

   • Under H0, X ~ B(25, 0.4).
   • P(X ≥ 14) = 0.0778 > 0.05
   • P(X ≥ 15) = 0.0344 < 0.05
   • Critical region: X ≥ 15

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

   • P(X ≥ 15) = 0.0344
   • The actual significance level is 3.44%.

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

   • Including 14 gives P(X ≥ 14) = 0.0778, 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 coin is tested for bias towards heads: H0: p = 0.5, H1: p > 0.5, at the 5% significance level. For each number of tosses n, find the critical region and its actual probability when H0 is true.

1. (a) n = 10

   • Under H0, X ~ B(10, 0.5).
   • P(X ≥ 8) = 0.0547 > 0.05
   • P(X ≥ 9) = 0.0107 < 0.05
   • Critical region: X ≥ 9, with probability 0.0107

2. (b) n = 20

   • Under H0, X ~ B(20, 0.5).
   • P(X ≥ 14) = 0.0577 > 0.05
   • P(X ≥ 15) = 0.0207 < 0.05
   • Critical region: X ≥ 15, with probability 0.0207

3. (c) n = 30

   • Under H0, X ~ B(30, 0.5).
   • P(X ≥ 19) = 0.100 > 0.05
   • P(X ≥ 20) = 0.0494 < 0.05
   • Critical region: X ≥ 20, with probability 0.0494