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Answer :
Final answer:
To find the probability that the mean weight of a sample of size 100 will be greater than 117 pounds, we use the Central Limit Theorem and calculate the z-score. The calculated probability is approximately 0.04%.
Explanation:
To find the probability that the mean weight of a sample of size 100 will be greater than 117 pounds, we need to use the Central Limit Theorem. The Central Limit Theorem states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases. In this case, the mean weight is 115 pounds with a standard deviation of 6 pounds. Since the sample size is 100, we can assume that the sampling distribution of the sample mean will be approximately normal.
To calculate the probability, we need to calculate the z-score and then find the corresponding probability using the z-table. The formula to calculate the z-score is:
z = (x - μ) / (σ / √n)
Where:
- z is the z-score
- x is the mean weight in question (117 pounds)
- μ is the population mean (115 pounds)
- σ is the population standard deviation (6 pounds)
- n is the sample size (100)
Plugging in the values:
z = (117 - 115) / (6 / √100) = 2 / 0.6 = 3.33
Using the z-table, we can find that the probability of getting a z-score greater than 3.33 is approximately 0.0004 (or 0.04%). Therefore, the probability that the mean weight of a sample of size 100 will be greater than 117 pounds is approximately 0.04%.
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