Question

Suppose x has a distribution with a mean of 90 and a standard deviation of 3. Random samples of size n = 36 are drawn. (a) Describe the x distribution and compute the mean and standard deviation of the distribution. x has a normal distribution with mean μx = 90 and standard deviation σx = 3. (b) Find the z value corresponding to x = 91. z = (91 - 90) / 3 = 0.3333. (c) Find P(x < 91). P(x < 91) = 0.6306. (d) Would it be unusual for a random sample of size 36 from the x distribution to have a sample mean less than 91? Explain. No, it would not be unusual because less than 5% of all such samples have means less than 91.

Answer 1

(a) The distribution of x is normal with a mean
`(\(\mu_x\))` of 90 and a standard deviation
`(\(\sigma_x\))` of 3.

(b) The z value corresponding to x = 91 is z = 0.3333.

(c) The probability P(x < 91) is 0.6306.

(d) It would not be unusual for a random sample of size 36 from the x distribution to have a sample mean less than 91, as less than 5% of all such samples have means less than 91.

Step-by-step explanation:

(a) The given information describes the distribution of x as normal with a mean (
`\(\mu_x\)`) of 90 and a standard deviation (
`\(\sigma_x\)`) of 3. This implies that x follows a bell-shaped curve centered around 90, and about 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.

(b) To find the z value corresponding to x = 91, the formula
`\(z = ((x - \mu_x))/(\sigma_x)\)` is used, where x is the specific value,
`\(\mu_x\)` is the mean, and
`\(\sigma_x\)` is the standard deviation. Substituting the given values, we find z = 0.3333.

(c) The probability
`\(P(x < 91)\)` is the area under the normal distribution curve to the left of x = 91. This probability can be found using standard normal distribution tables or statistical software, resulting in P(x < 91) = 0.6306.

(d) The statement that it would not be unusual for a random sample of size 36 to have a sample mean less than 91 is based on the fact that, in a normal distribution, about 95% of sample means from random samples of size 36 will fall within 1.96 standard deviations of the population mean. Since the mean is 90 and the standard deviation is 3, a mean of 91 falls within this range, making it a common occurrence in the distribution of sample means.