un times at a local marathon are known to follow a left-skewed distribution with a mean of 256 minutes and a standard deviation of 100 minutes. If we select a random sample of 46 people, what is the probability that the average run time of this sample will be between 231 minutes and 264 minutes?Give your answer to four decimal places. If the question cannot be answered with the information given, enter -1 as your answer.
Question
un times at a local marathon are known to follow a left-skewed distribution with a mean of 256 minutes and a standard deviation of 100 minutes. If we select a random sample of 46 people, what is the probability that the average run time of this sample will be between 231 minutes and 264 minutes?Give your answer to four decimal places. If the question cannot be answered with the information given, enter -1 as your answer.
Solution
To solve this problem, we will use the Central Limit Theorem which states that the sampling distribution of the sample means approaches a normal distribution as the sample size gets larger, regardless of the shape of the population distribution. This theorem applies when the sample size is larger than 30, which is the case here (46 > 30).
Step 1: Identify the given information
- The population mean (μ) is 256 minutes.
- The population standard deviation (σ) is 100 minutes.
- The sample size (n) is 46.
- We are looking for the probability that the sample mean (x̄) is between 231 and 264 minutes.
Step 2: Calculate the standard error of the mean (SE) SE = σ / sqrt(n) = 100 / sqrt(46) ≈ 14.73
Step 3: Convert the sample mean range to z-scores The z-score is calculated as (x̄ - μ) / SE.
For 231 minutes: z1 = (231 - 256) / 14.73 ≈ -1.70
For 264 minutes: z2 = (264 - 256) / 14.73 ≈ 0.54
Step 4: Find the probabilities corresponding to these z-scores You can use a standard normal distribution table or a calculator to find these probabilities.
P(z < -1.70) ≈ 0.0446 P(z < 0.54) ≈ 0.7054
Step 5: Calculate the probability that the sample mean is between 231 and 264 minutes P(231 < x̄ < 264) = P(z < 0.54) - P(z < -1.70) = 0.7054 - 0.0446 = 0.6608
So, the probability that the average run time of a random sample of 46 people will be between 231 minutes and 264 minutes is approximately 0.6608 or 66.08%.
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