This section focuses on Z-tests for population means (where the population standard deviation is known) and Z-tests for population proportions. These questions will guide you through the necessary calculations.
A coffee machine is designed to dispense an average of 240 ml of coffee per cup. The manufacturer knows from past experience that the amount of coffee dispensed is normally distributed with a population standard deviation (σ) of 10 ml. A random sample of 36 cups is taken, and the sample mean (x̄) is found to be 236 ml. Test the hypothesis at a 5% significance level (α = 0.05) that the machine is under-filling the cups (i.e., the true mean is less than 240 ml).
For calculations involving the normal distribution, you might find this tool helpful:
Step 1: State the Hypotheses
This is a left-tailed test.
Step 2: Significance Level
α = 0.05
Step 3: Calculate the Test Statistic
Formula: Z = (x̄ - μ₀) / (σ / √n)
Where: x̄ = 236 ml, μ₀ = 240 ml, σ = 10 ml, n = 36
Z = (236 - 240) / (10 / √36) = -4 / (10 / 6) = -4 / 1.6667 = -2.40
Step 4: Determine the Critical Value / P-value
Critical Value Approach: For a left-tailed test with α = 0.05, critical Z-value ≈ -1.645. Rejection region: Z < -1.645.
P-value Approach: P-value = P(Z < -2.40) ≈ 0.0082.
Step 5: Make a Decision
Calculated Z (-2.40) < Critical Z (-1.645). OR P-value (0.0082) < α (0.05).
Step 6: Conclusion
Reject H₀. There is sufficient evidence at the 5% significance level to conclude that the machine is under-filling the cups.
A car manufacturer claims that its new model has an average fuel efficiency of more than 30 miles per gallon (mpg). The population standard deviation for fuel efficiency is known to be 2.5 mpg. A random sample of 49 cars of this model showed a sample mean fuel efficiency of 31.2 mpg. Test the manufacturer's claim at a 1% significance level (α = 0.01).
Step 1: State the Hypotheses
This is a right-tailed test.
Step 2: Significance Level
α = 0.01
Step 3: Calculate the Test Statistic
Formula: Z = (x̄ - μ₀) / (σ / √n)
Where: x̄ = 31.2 mpg, μ₀ = 30 mpg, σ = 2.5 mpg, n = 49
Z = (31.2 - 30) / (2.5 / √49) = 1.2 / (2.5 / 7) = 1.2 / 0.3571 ≈ 3.36
Step 4: Determine the Critical Value / P-value
Critical Value Approach: For a right-tailed test with α = 0.01, critical Z-value ≈ 2.326. Rejection region: Z > 2.326.
P-value Approach: P-value = P(Z > 3.36) = 1 - P(Z ≤ 3.36) ≈ 1 - 0.9996 = 0.0004.
Step 5: Make a Decision
Calculated Z (3.36) > Critical Z (2.326). OR P-value (0.0004) < α (0.01).
Step 6: Conclusion
Reject H₀. There is sufficient evidence at the 1% significance level to support the manufacturer's claim that the average fuel efficiency is more than 30 mpg.
A politician claims that more than 60% of their constituents support a new bill. In a random sample of 500 constituents, 315 said they support the bill. Test the politician's claim at a 5% significance level (α = 0.05).
Step 1: State the Hypotheses
This is a right-tailed test.
Step 2: Significance Level
α = 0.05
Step 3: Calculate the Sample Proportion (p̂) and Test Statistic (Z)
Sample proportion (p̂) = x / n = 315 / 500 = 0.63
Check conditions: np₀ = 500 * 0.60 = 300 ≥ 10; n(1-p₀) = 500 * 0.40 = 200 ≥ 10. Conditions met.
Formula: Z = (p̂ - p₀) / √[p₀(1-p₀)/n]
Z = (0.63 - 0.60) / √[0.60(0.40)/500] = 0.03 / √0.00048 ≈ 1.37
Step 4: Determine the Critical Value / P-value
Critical Value Approach: For α = 0.05 (right-tailed), critical Z-value = 1.645. Rejection region: Z > 1.645.
P-value Approach: P-value = P(Z > 1.37) ≈ 0.0853.
Step 5: Make a Decision
Calculated Z (1.37) is not > Critical Z (1.645). OR P-value (0.0853) > α (0.05).
Step 6: Conclusion
Fail to reject H₀. There is not sufficient evidence at the 5% significance level to support the politician's claim.
The average weight of a certain brand of cereal is stated to be 500 grams. The population standard deviation is known to be 15 grams. A quality control inspector takes a sample of 49 boxes and finds the sample mean weight to be 494 grams. Is there evidence at the 5% significance level that the true mean weight is different from 500 grams?
Step 1: State the Hypotheses
This is a two-tailed test.
Step 2: Significance Level
α = 0.05
Step 3: Calculate the Test Statistic
Z = (x̄ - μ₀) / (σ / √n)
Z = (494 - 500) / (15 / √49) = -6 / (15 / 7) ≈ -2.80
Step 4: Determine the Critical Values / P-value
Critical Value Approach: For α = 0.05 (two-tailed), critical Z-values = ±1.96. Rejection region: Z < -1.96 or Z > 1.96.
P-value Approach: P-value = 2 * P(Z < -2.80) ≈ 2 * 0.0026 = 0.0052.
Step 5: Make a Decision
Calculated Z (-2.80) < -1.96. OR P-value (0.0052) < α (0.05).
Step 6: Conclusion
Reject H₀. There is sufficient evidence at the 5% significance level that the true mean weight of the cereal boxes is different from 500 grams.
A local health department claims that the proportion of adults in their city who smoke is 20% (p = 0.20). A critic believes this proportion is actually lower. A random sample of 400 adults is surveyed, and 70 are found to be smokers. Test the critic's belief at a 10% significance level (α = 0.10).
Step 1: State the Hypotheses
This is a left-tailed test.
Step 2: Significance Level
α = 0.10
Step 3: Calculate the Sample Proportion (p̂) and Test Statistic (Z)
Sample proportion (p̂) = x / n = 70 / 400 = 0.175
Check conditions: np₀ = 400 * 0.20 = 80 ≥ 10; n(1-p₀) = 400 * 0.80 = 320 ≥ 10. Conditions met.
Formula: Z = (p̂ - p₀) / √[p₀(1-p₀)/n]
Z = (0.175 - 0.20) / √[0.20(0.80)/400] = -0.025 / √[0.16/400] = -0.025 / √0.0004 = -0.025 / 0.02 = -1.25
Step 4: Determine the Critical Value / P-value
Critical Value Approach: For α = 0.10 (left-tailed), critical Z-value ≈ -1.282. Rejection region: Z < -1.282.
P-value Approach: P-value = P(Z < -1.25) ≈ 0.1056.
Step 5: Make a Decision
Calculated Z (-1.25) is not < Critical Z (-1.282). OR P-value (0.1056) > α (0.10).
Step 6: Conclusion
Fail to reject H₀. There is not sufficient evidence at the 10% significance level to support the critic's belief that the proportion of smokers is lower than 20%.
A survey was conducted to compare the proportion of male and female college students who own a car. Males: n₁ = 200, x₁ (own car) = 120. Females: n₂ = 250, x₂ (own car) = 130. Test at α = 0.01 if there is a significant difference in the proportion of male and female students who own a car.
Step 1: Calculate Sample Proportions (p̂₁, p̂₂) and Pooled Proportion (p̂pool), State Hypotheses
p̂₁ (males) = 120 / 200 = 0.60. p̂₂ (females) = 130 / 250 = 0.52
Pooled proportion (p̂pool) = (x₁ + x₂) / (n₁ + n₂) = (120 + 130) / (200 + 250) = 250 / 450 ≈ 0.5556
This is a two-tailed test.
Check conditions for Z-test: n₁p̂pool, n₁(1-p̂pool), n₂p̂pool, n₂(1-p̂pool) are all ≥ 5 (or 10). 200*0.5556 ≈ 111; 200*0.4444 ≈ 89; 250*0.5556 ≈ 139; 250*0.4444 ≈ 111. All are ≥ 10. Conditions met.
Step 2: Significance Level
α = 0.01
Step 3: Calculate the Test Statistic (Z)
Formula: Z = (p̂₁ - p̂₂) / √[p̂pool(1-p̂pool)(1/n₁ + 1/n₂)]
Z = (0.60 - 0.52) / √[0.5556(0.4444)(1/200 + 1/250)] = 0.08 / √[0.2469(0.005 + 0.004)] = 0.08 / √0.0022221 ≈ 1.697
Step 4: Determine the Critical Values / P-value
Critical Value Approach: For α = 0.01 (two-tailed), critical Z-values ≈ ±2.576. Rejection region: Z < -2.576 or Z > 2.576.
P-value Approach: P-value = 2 * P(Z > |1.697|) ≈ 2 * (1 - 0.9551) = 0.0898.
Step 5: Make a Decision
Calculated Z (1.697) is between -2.576 and 2.576. OR P-value (0.0898) > α (0.01).
Step 6: Conclusion
Fail to reject H₀. There is not sufficient evidence at the 1% significance level to conclude a difference in car ownership proportions.
A machine produces bolts with a target mean diameter of 10.00 mm. The population standard deviation of diameters is 0.02 mm. A sample of 100 bolts has a sample mean diameter of 10.005 mm. Does this provide evidence at α = 0.05 that the machine is not producing bolts with the target mean diameter?
Step 1: State the Hypotheses
This is a two-tailed test.
Step 2: Significance Level
α = 0.05
Step 3: Calculate the Test Statistic (Z)
Z = (x̄ - μ₀) / (σ / √n)
Z = (10.005 - 10.00) / (0.02 / √100) = 0.005 / (0.02 / 10) = 0.005 / 0.002 = 2.50
Step 4: Determine the Critical Values / P-value
Critical Value Approach: For α = 0.05 (two-tailed), critical Z-values = ±1.96. Rejection region: Z < -1.96 or Z > 1.96.
P-value Approach: P-value = 2 * P(Z > 2.50) = 2 * (1 - P(Z ≤ 2.50)) ≈ 2 * (1 - 0.9938) = 2 * 0.0062 = 0.0124.
Step 5: Make a Decision
Calculated Z (2.50) > 1.96. OR P-value (0.0124) < α (0.05).
Step 6: Conclusion
Reject H₀. There is sufficient evidence at the 5% significance level that the machine is not producing bolts with the target mean diameter of 10.00 mm.
A company claims that 30% of its website visitors make a purchase. In a sample of 1000 visitors, 270 made a purchase. Test at α = 0.10 if the true proportion of visitors making a purchase is different from 30%.
Step 1: State the Hypotheses
This is a two-tailed test.
Step 2: Significance Level
α = 0.10
Step 3: Calculate the Sample Proportion (p̂) and Test Statistic (Z)
p̂ = 270 / 1000 = 0.27
Check conditions: np₀ = 1000*0.30 = 300 ≥ 10; n(1-p₀) = 1000*0.70 = 700 ≥ 10. Conditions met.
Z = (p̂ - p₀) / √[p₀(1-p₀)/n]
Z = (0.27 - 0.30) / √[0.30(0.70)/1000] = -0.03 / √[0.21/1000] = -0.03 / √0.00021 = -0.03 / 0.01449 ≈ -2.07
Step 4: Determine the Critical Values / P-value
Critical Value Approach: For α = 0.10 (two-tailed), critical Z-values ≈ ±1.645. Rejection region: Z < -1.645 or Z > 1.645.
P-value Approach: P-value = 2 * P(Z < -2.07) ≈ 2 * 0.0192 = 0.0384.
Step 5: Make a Decision
Calculated Z (-2.07) < -1.645. OR P-value (0.0384) < α (0.10).
Step 6: Conclusion
Reject H₀. There is sufficient evidence at the 10% significance level that the true proportion of visitors making a purchase is different from 30%.
A study compares the mean IQ scores of students from two different schools. School A: n₁ = 50, x̄₁ = 105, σ₁ (population SD for School A) = 15. School B: n₂ = 60, x̄₂ = 102, σ₂ (population SD for School B) = 12. Test at α = 0.05 if there is a significant difference in the mean IQ scores between the two schools.
Step 1: State the Hypotheses
This is a two-tailed test.
Step 2: Significance Level
α = 0.05
Step 3: Calculate the Test Statistic (Z)
Formula: Z = (x̄₁ - x̄₂) / √[(σ₁²/n₁) + (σ₂²/n₂)]
Z = (105 - 102) / √[(15²/50) + (12²/60)]
Z = 3 / √[(225/50) + (144/60)] = 3 / √[4.5 + 2.4] = 3 / √6.9 = 3 / 2.62679 ≈ 1.142
Step 4: Determine the Critical Values / P-value
Critical Value Approach: For α = 0.05 (two-tailed), critical Z-values = ±1.96. Rejection region: Z < -1.96 or Z > 1.96.
P-value Approach: P-value = 2 * P(Z > 1.142) = 2 * (1 - P(Z ≤ 1.142)) ≈ 2 * (1 - 0.8733) = 2 * 0.1267 = 0.2534.
Step 5: Make a Decision
Calculated Z (1.142) is between -1.96 and 1.96. OR P-value (0.2534) > α (0.05).
Step 6: Conclusion
Fail to reject H₀. There is not sufficient evidence at the 5% significance level to conclude there is a significant difference in mean IQ scores between the two schools.
A pharmaceutical company develops a new drug (Drug X) and wants to compare its effectiveness (proportion of patients cured) against an existing drug (Drug Y). Drug X: n₁ = 300 patients, x₁ (cured) = 210. Drug Y: n₂ = 250 patients, x₂ (cured) = 150. Test at α = 0.05 if Drug X is more effective than Drug Y.
Step 1: Calculate Sample Proportions and Pooled Proportion, State Hypotheses
p̂₁ (Drug X) = 210 / 300 = 0.70
p̂₂ (Drug Y) = 150 / 250 = 0.60
Pooled proportion (p̂pool) = (x₁ + x₂) / (n₁ + n₂) = (210 + 150) / (300 + 250) = 360 / 550 ≈ 0.6545
This is a right-tailed test.
Check conditions for Z-test (as in Q6).
Step 2: Significance Level
α = 0.05
Step 3: Calculate the Test Statistic (Z)
Formula: Z = (p̂₁ - p̂₂) / √[p̂pool(1-p̂pool)(1/n₁ + 1/n₂)]
Z = (0.70 - 0.60) / √[0.6545(1-0.6545)(1/300 + 1/250)]
Z = 0.10 / √[0.6545(0.3455)(0.003333 + 0.004)]
Z = 0.10 / √[0.22619(0.007333)] = 0.10 / √0.001659 ≈ 0.10 / 0.04073 ≈ 2.455
Step 4: Determine the Critical Value / P-value
Critical Value Approach: For α = 0.05 (right-tailed), critical Z-value = 1.645. Rejection region: Z > 1.645.
P-value Approach: P-value = P(Z > 2.455) = 1 - P(Z ≤ 2.455) ≈ 1 - 0.9929 = 0.0071.
Step 5: Make a Decision
Calculated Z (2.455) > Critical Z (1.645). OR P-value (0.0071) < α (0.05).
Step 6: Conclusion
Reject H₀. There is sufficient evidence at the 5% significance level to conclude that Drug X is more effective than Drug Y.
In a left-tailed Z-test for a population mean (σ known), if the calculated Z-statistic is -2.00 and the critical Z-value is -1.96 at α = 0.05, what is the decision?
If the p-value for a two-tailed Z-test is 0.065 and the significance level (α) is 0.05, what is the conclusion?
A Z-test for a population proportion is appropriate when:
In a Z-test for the difference between two population means (σ₁ and σ₂ known), if the confidence interval for μ₁ - μ₂ is (-0.5, 2.5), what would be the decision for a two-tailed test of H₀: μ₁ - μ₂ = 0 at the corresponding α level?
If we decrease the significance level (α) from 0.05 to 0.01, what happens to the probability of making a Type I error and the critical value(s) for a Z-test?