Question

ID. NRG Sec s erial #: 3. A small business ships specialty homemade candies to anywhere in the world. Past records indicate that orders weigh an average of 118 grams with a standard deviation of 14 grams. Suppose a random sample of 50 current orders is selected and each weighed. a. Describe the sampling distribution for the sample mean by naming the model and telling its mean and standard deviation. b. Suppose the resulting sample mean is 110 grams. Do you think that this sample result is unusually small? Explain.

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Answer 1

Solution:

Given:

Mean = $\mu = 118$

Standard Deviation = $\sigma = 14$

Sample size = n = 50

Part a) sampling distribution of sample mean:

Since sample size n = 50 is large , we can use Central limit theorem which states that for large sample size n ,
sampling distribution of sample mean is approximately normal with mean of sample means:

$\mu_{\bar{x}}=\mu$

$\mu_{\bar{x}}=118$

and standard deviation of sample means is:

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$\sigma_{\bar{x}}= \frac{14}{\sqrt{50}}$

$\sigma_{\bar{x}}= \frac{14}{7.071068 }$

$\sigma_{\bar{x}}=1.98$

Part b)

Sample mean = $\bar{x} = 110$

We have to determine if this sample mean result is unusually small or not.

Thus find find z score for $\bar{x} = 110$ :

$z=\frac{\bar{x}-\mu_{\bar{x}} }{\sigma_{\bar{x}}}$

$z=\frac{110-118 }{1.98}$

$z=\frac{-8 }{1.98}$

$z=-4.04$

Since this z score is less than z = -2.00, the sample result is unusually small.