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10. L-13 Points] DETAILS PODSTAT6 8.E.033.S. MY NOTES ASK YOUR TEACHER Let *, *...*100 denote the actual net weights (in pounds) of 100 randomly selected bags of fertilizer. Suppose that the weight of a randomly selected bag has a distribution with mean 40 lbs and variance 1 tb? Let X be the sample mean weight (n = 100) USE SALT (a) Describe the sampling distribution of x. The distribution is unknown with unknown mean and variance. The distribution is approximately normal with a mean of 40 lbs and variance of 0.01 lbs? The distribution is unknown with a mean of 40 lbs and variance of 0.01 lbs? The distribution is unknown with a mean of 40 lbs and variance of 1 1b2 The distribution is approximately normal with a mean of 40 lbs and variance of 1 1b2 (b) What is the probability that the sample mean is between 39.85 lbs and 40.15 lbs? (Round your answer to four decimal places.) P(39.85 s * s 40.15) (c) What is the probability that the sample mean is greater than 40 lbs?

Answer :

a) The distribution is approximately normal with a mean of 40 lbs and variance of 0.01 lbs.

b) The probability that the sample mean is between 39.85 lbs and 40.15 lbs is approximately 0.8664.

c) The probability that the sample mean is greater than 40 lbs is approximately 0.5.

a) The central limit theorem states that for a sufficiently large sample size (n = 100 in this case), the distribution of the sample mean approaches a normal distribution, regardless of the underlying population distribution. The mean of the sampling distribution of the sample mean is equal to the population mean (40 lbs), and the variance of the sampling distribution is equal to the population variance divided by the sample size (1 lbs² / 100 = 0.01 lbs²).

b) To find the probability that the sample mean falls between 39.85 lbs and 40.15 lbs, we can use the properties of the normal distribution. We need to calculate the area under the normal distribution curve between these two values. This probability can be found using standard normal distribution tables or a calculator and is approximately 0.8664.

c) The probability that the sample mean is greater than 40 lbs can be calculated by finding the area under the normal distribution curve to the right of 40 lbs. This probability is 0.5, as the normal distribution is symmetric around its mean, and half of the area lies to the right of the mean.

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