Use the central limit theorem to find the approximate probability that the mean number of flaws exceeds 1.6 per square yard. (Round your answer to four decimal places.)
I bet there is considerably more to this question.
To use the central limit theorem to find the approximate probability that the mean number of flaws exceeds 1.6 per square yard, you need to follow these steps:
1. Identify the distribution: Start by knowing the distribution of the random variable you are dealing with. In this case, you need to know the distribution of the number of flaws per square yard.
2. Identify the sample size: Determine the sample size of the data you have. This is important for the central limit theorem to be applicable. Typically, a sample size of at least 30 is considered sufficient.
3. Calculate the mean and standard deviation: Find the mean (μ) and standard deviation (σ) of the distribution of the number of flaws per square yard.
4. Establish the sampling distribution: Since sample means are the focus of the central limit theorem, you can consider the distribution of sample means. This distribution approximates a normal distribution, regardless of the shape of the original distribution, as long as the sample size is large enough.
5. Calculate the standard error: The standard error (SE) represents the standard deviation of the sampling distribution. It is calculated by dividing the standard deviation of the original distribution (σ) by the square root of the sample size (n).
6. Standardize the sample mean: To proceed with finding the probability, standardize your sample mean using the formula: z = (x - μ) / SE. Here, x is the value you are interested in (1.6), μ is the mean of the original distribution, and SE is the standard error calculated in step 5.
7. Find the probability: Use a standard normal distribution table or a calculator to find the probability associated with the standardized value calculated in step 6. Remember to consider whether you want the probability of the mean number of flaws exceeding 1.6 or the probability of it being less than or equal to 1.6, depending on the wording of the question.
By following these steps, you should be able to approximate the probability that the mean number of flaws exceeds 1.6 per square yard using the central limit theorem. Round the answer to four decimal places as specified in the question.