Categorical Data

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Answer Key & Explanations

Categorical Data · Grade 11 · Worksheet 3

1. Aroha surveyed students about their favorite sport. Results: 15 soccer, 11 basketball, 9 rugby, 13 cricket, 7 tennis. Create a frequency table and find the total number of students surveyed. Answer: 55 Solution: Soccer: 15 Basketball: 11 Rugby: 9 Cricket: 13 Tennis: 7 15 + 11 = 26 26 + 9 = 35 35 + 13 = 48 48 + 7 = 55 The total number of students surveyed is 55.
   Full step-by-step solution

   Step 1: List each sport and its frequency: Soccer: 15 Basketball: 11 Rugby: 9 Cricket: 13 Tennis: 7 Step 2: Add all frequencies to find the total: 15 + 11 = 26 26 + 9 = 35 35 + 13 = 48 48 + 7 = 55 The total number of students surveyed is 55.

2. A pharmaceutical company is testing a new drug and claims it reduces recovery time from a certain illness. In a clinical trial with 120 patients, the mean recovery time was 8.2 days with a standard deviation of 1.5 days. The company wants to test the hypothesis that the mean recovery time is less than the standard treatment's 9 days at a 5% significance level. Calculate the test statistic for this hypothesis test and determine if there is sufficient evidence to reject the null hypothesis. Answer: z ≈ -5.84, reject H₀ Solution: State the hypotheses. We are testing if the new drug reduces recovery time compared to the standard treatment's 9 days.
   Full step-by-step solution

   Step 1: State the hypotheses. We are testing if the new drug reduces recovery time compared to the standard treatment's 9 days. Null hypothesis H₀: μ = 9 (mean recovery time is 9 days) Alternative hypothesis H₁: μ < 9 (mean recovery time is less than 9 days) This is a left-tailed test. Step 2: Identify given data. Sample size n = 120 Sample mean x̄ = 8.2 Population standard deviation σ = 1.5 (we use the sample standard deviation as an estimate for σ since n is large) Population mean under H₀ μ₀ = 9 Significance level α = 0.05 Step 3: Choose the test statistic. Because n > 30, we use the z-test for means. The test statistic formula is: z = (x̄ - μ₀) / (σ / sqrt(n)) Step 4: Calculate the standard error. Standard error = σ / sqrt(n) = 1.5 / sqrt(120) First, sqrt(120) ≈ 10.954451 So standard error = 1.5 / 10.954451 ≈ 0.136931 Step 5: Calculate the z-test statistic. z = (8.2 - 9) / 0.136931 z = (-0.8) / 0.136931 z ≈ -5.84 Step 6: Determine the critical value and compare. For α = 0.05 in a left-tailed test, the critical z-value is -1.645. Our calculated z ≈ -5.84 is much less than -1.645. Step 7: Make a decision. Since the test statistic is in the rejection region (z < -1.645), we reject H₀. Step 8: Conclusion. There is sufficient evidence at the 5% significance level to reject the null hypothesis. The data supports the claim that the new drug reduces recovery time compared to the standard treatment. Final answer: z ≈ -5.84, reject H₀

3. A pharmaceutical company is testing a new drug and claims it reduces recovery time from a certain illness. In a clinical trial with 250 patients, the mean recovery time was 38.2 hours with a standard deviation of 6.5 hours. The company wants to test if this is significantly different from the standard treatment's mean recovery time of 40 hours at a 5% significance level. What is the calculated test statistic for this hypothesis test? Answer: -4.38 Solution: We are comparing the sample mean (38.2 hours) to a known population mean (40 hours) for the standard treatment.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Identify the type of test** We are comparing the sample mean (38.2 hours) to a known population mean (40 hours) for the standard treatment. We know: - Sample size \( n = 250 \) - Sample mean \( \bar{x} = 38.2 \) - Sample standard deviation \( s = 6.5 \) - Population mean under null hypothesis \( \mu_0 = 40 \) Since \( n \) is large (\( n > 30 \)), we use a **z-test** (though some might use t-test, but z is fine here because of large n). --- **Step 2: State the hypotheses** Null hypothesis \( H_0: \mu = 40 \) Alternative hypothesis \( H_a: \mu \neq 40 \) (two-tailed test) Significance level \( \alpha = 0.05 \) --- **Step 3: Formula for the test statistic** For a one-sample z-test: \[ z = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} \] --- **Step 4: Plug in the numbers** \[ z = \frac{38.2 - 40}{6.5 / \sqrt{250}} \] First, compute the denominator: \[ \sqrt{250} \approx 15.811388 \] \[ 6.5 / 15.811388 \approx 0.411096 \] Now numerator: \[ 38.2 - 40 = -1.8 \] Now divide: \[ z = \frac{-1.8}{0.411096} \approx -4.378 \] --- **Step 5: Rounding** Rounded to two decimal places: \[ z \approx -4.38 \] --- **Final Answer:** The calculated test statistic is **-4.38**.

4. A medical researcher is testing a new drug's effect on cholesterol levels. In a clinical trial with 180 patients, the mean reduction in LDL cholesterol was 24.3 mg/dL with a standard deviation of 8.7 mg/dL. The researcher wants to construct a 99% confidence interval for the true mean reduction. Calculate the margin of error for this confidence interval, rounding your answer to two decimal places. Answer: 1.67 Solution: Sample size n = 180 Sample mean = 24.3 mg/dL (not needed for margin of error) Sample standard deviation s = 8.7 mg/dL Confidence level = 99% Find the critical value for 99% confidence For a large sample (n > 30), we use the z-distribution z-value for 99% confidence is 2.576 Standard error = s /…
   Full step-by-step solution

   Step 1: Identify the known values Sample size n = 180 Sample mean = 24.3 mg/dL (not needed for margin of error) Sample standard deviation s = 8.7 mg/dL Confidence level = 99% Step 2: Find the critical value for 99% confidence For a large sample (n > 30), we use the z-distribution z-value for 99% confidence is 2.576 Step 3: Calculate the standard error Standard error = s / sqrt(n) = 8.7 / sqrt(180) sqrt(180) ≈ 13.4164 Standard error = 8.7 / 13.4164 ≈ 0.6484 Step 4: Calculate the margin of error Margin of error = z-value × standard error Margin of error = 2.576 × 0.6484 ≈ 1.670 Step 5: Round to two decimal places Margin of error ≈ 1.67 The margin of error for the 99% confidence interval is 1.67 mg/dL.

5. A medical researcher is conducting a clinical trial for a new migraine medication. In a sample of 80 patients, the mean reduction in migraine frequency was 4.2 migraines per month with a standard deviation of 1.8 migraines. The researcher wants to test if this reduction is significantly different from the standard medication's average reduction of 3.5 migraines per month. Using a significance level of α = 0.01, calculate the test statistic for this hypothesis test. Answer: 3.48 Solution: H₀: μ = 3.5 (no difference from standard medication) H₁: μ ≠ 3.5 (different from standard medication) Sample mean (x̄) = 4.2 Population mean (μ) = 3.5 Sample standard deviation (s) = 1.8 Sample size (n) = 80 Standard error = s/√n = 1.8/√80 = 1.8/8.944 = 0.201 t = (x̄ - μ)/(s/√n) = (4.2 -…
   Full step-by-step solution

   Step 1: Identify the null and alternative hypotheses H₀: μ = 3.5 (no difference from standard medication) H₁: μ ≠ 3.5 (different from standard medication) Step 2: Identify the sample statistics Sample mean (x̄) = 4.2 Population mean (μ) = 3.5 Sample standard deviation (s) = 1.8 Sample size (n) = 80 Step 3: Calculate the standard error Standard error = s/√n = 1.8/√80 = 1.8/8.944 = 0.201 Step 4: Calculate the test statistic t = (x̄ - μ)/(s/√n) = (4.2 - 3.5)/0.201 = 0.7/0.201 = 3.48 The test statistic is 3.48.

6. Hana surveyed 63 students about their favorite type of book. 24 preferred Fiction, 18 preferred Non-Fiction, 12 preferred Mystery, and 9 preferred Fantasy. Create a frequency table for this categorical data. Answer: Fiction: 24, Non-Fiction: 18, Mystery: 12, Fantasy: 9 Solution: List all categories: Fiction, Non-Fiction, Mystery, Fantasy. Fiction: 24 Non-Fiction: 18 Mystery: 12 Fantasy: 9 The answer is Fiction: 24, Non-Fiction: 18, Mystery: 12, Fantasy: 9.
   Full step-by-step solution

   Step 1: List all categories: Fiction, Non-Fiction, Mystery, Fantasy. Step 2: Record the frequency for each category: - Fiction: 24 students - Non-Fiction: 18 students - Mystery: 12 students - Fantasy: 9 students Step 3: Verify the total: 24 + 18 + 12 + 9 = 63 students. Step 4: The completed frequency table is: Fiction: 24 Non-Fiction: 18 Mystery: 12 Fantasy: 9 The answer is Fiction: 24, Non-Fiction: 18, Mystery: 12, Fantasy: 9.