SOLUTION: A study of 243 randomly selected occupants in passenger cars and 199 randomly selected occupants in pickup trucks shows that 91% of occupants in passenger cars and 88% of occupants

Question 1179001: A study of 243 randomly selected occupants in passenger cars and 199 randomly selected occupants in pickup trucks shows that 91% of occupants in passenger cars and 88% of occupants in pickup trucks wear seat belts. At α=0.10, can you reject the claim that the proportion of occupants who wear seat belts is the different for passenger cars and pickup trucks?

Conditions:
Significant:
Hypothesis:
Conclusion:
Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
Here's how to conduct a hypothesis test to determine if the proportions of seatbelt wearers in passenger cars and pickup trucks are different.
**1. Conditions:**
* **Random Samples:** Given (243 car occupants, 199 truck occupants)
* **Independence:** Assumed (samples are independent of each other)
* **Normality (Large Samples):**
* n₁p₁ = 243 * 0.91 = 221.13 > 10
* n₁(1-p₁) = 243 * 0.09 = 21.87 > 10
* n₂p₂ = 199 * 0.88 = 175.12 > 10
* n₂(1-p₂) = 199 * 0.12 = 23.88 > 10
* All conditions are met.
**2. Significance Level:**
* α = 0.10
**3. Hypothesis:**
* **Null Hypothesis (H₀):** p₁ = p₂ (The proportions are equal)
* **Alternative Hypothesis (H₁):** p₁ ≠ p₂ (The proportions are different)
* This is a two-tailed test.
**4. Calculate Sample Proportions:**
* p̂₁ (car) = 0.91
* p̂₂ (truck) = 0.88
* n₁ (car) = 243
* n₂ (truck) = 199
**5. Calculate the Pooled Proportion (p̂):**
* p̂ = (x₁ + x₂) / (n₁ + n₂) = (n₁p̂₁ + n₂p̂₂) / (n₁ + n₂)
* p̂ = (243 * 0.91 + 199 * 0.88) / (243 + 199)
* p̂ = (221.13 + 175.12) / 442
* p̂ = 396.25 / 442 ≈ 0.8965
**6. Calculate the Standard Error (SE):**
* SE = √[p̂(1 - p̂)(1/n₁ + 1/n₂)]
* SE = √[0.8965(1 - 0.8965)(1/243 + 1/199)]
* SE = √[0.8965(0.1035)(0.004115 + 0.005025)]
* SE = √[0.09278775(0.00914)]
* SE = √0.00084795
* SE ≈ 0.0291
**7. Calculate the Test Statistic (z-score):**
* z = (p̂₁ - p̂₂) / SE
* z = (0.91 - 0.88) / 0.0291
* z = 0.03 / 0.0291
* z ≈ 1.031
**8. Find the Critical Value:**
* For a two-tailed test with α = 0.10, the critical z-values are ±1.645.
**9. Make a Decision:**
* Compare the calculated z-score (1.031) to the critical z-values (±1.645).
* Since -1.645 < 1.031 < 1.645, the calculated z-score does not fall in the rejection region.
* Therefore, we fail to reject the null hypothesis.
**10. Conclusion:**
* There is not sufficient evidence at the 0.10 significance level to conclude that the proportion of occupants who wear seat belts is different for passenger cars and pickup trucks.

RELATED QUESTIONS
Records of 40 used passenger cars and 40 used pickup trucks (none used commercially) were (answered by CPhill)

According to R.L. Polk & Co., the average age of cars and light trucks in the U.S. in... (answered by stanbon)

1. A Science teacher used two different languages in teaching her two classes. Thirty... (answered by CPhill)

The following data represent the number of traffic fatalities in Country A in 2015 by... (answered by Boreal)

In a room with 35 men, 80% of the occupants are women. How many women are in the... (answered by scott8148,jim_thompson5910,actuary)

In a room with 35 men, 80% of the occupants are women. How many women are in the... (answered by edjones)

Urgent help needed. please help!! NOTE: Answers using z-scores rounded to 3 (or more)... (answered by Boreal)

A study conducted at a certain high school shows that 61% of its graduates enroll in a... (answered by stanbon)

Question: Given 8 cars and 9 people, what is the probability that a randomly selected... (answered by stanbon)