a report says 65% of vechiles sold were SUV's.
A random sample of 100 vechiles sold in last month is taken.
Use the central limit theorem, find the probability that most 70% of the vechiles in the sample were SUV's
To find the probability that the percentage of SUVs in the sample is less than or equal to 70%, we can use the Central Limit Theorem.
The Central Limit Theorem states that for a large enough sample size, the sampling distribution of the sample mean (or in this case, sample proportion) will be approximately normally distributed, regardless of the shape of the population distribution.
In this case, we are looking for the probability that most 70% of the vehicles in the sample were SUVs, which means we are interested in the cumulative probability from 0% to 70%.
To calculate this probability, we need to convert the sample proportion to a standardized z-score. The formula for calculating the z-score is as follows:
z = (X - μ) / (σ / √n)
Where:
- X represents the sample proportion (in this case, 70%)
- μ represents the population proportion (in this case, 65%)
- σ represents the standard deviation of the population proportion (calculated as √(p * (1-p) / n), where p is the population proportion)
- n represents the sample size
Given that the report states 65% of vehicles sold were SUVs, we can calculate the standard deviation as follows:
σ = √(p * (1-p) / n) = √(0.65 * 0.35 / 100) ≈ 0.0486
Now, we can calculate the z-score:
z = (0.70 - 0.65) / (0.0486 / √100) ≈ 1.025
To find the probability associated with this z-score, we can consult a standard normal distribution table or use statistical software. The cumulative probability associated with a z-score of 1.025 is approximately 0.8431.
Therefore, the probability that most 70% of the vehicles in the sample were SUVs is approximately 0.8431.