Question

You want to see if a redesign of the cover of a mail-order catalog will increase sales. A very large number of customers will receive the original catalog, and a random sample of customers will receive the one with the new cover. For planning purposes, you are willing to assume that the sales from the new catalog will be approximately normal with $\sigma = 65$ dollars and that the mean for the original catalog will be $\mu = 40$ dollars. You decide to use a sample size of $n = 990$. You wish to test the following hypotheses. $H_0: \mu \le 40$ $H_a: \mu > 40$ You decide to reject $H_0$ if $\bar{x} > 43.08$ and to accept $H_0$ otherwise. (a) Find the probability of a Type I error, that is, the probability that your test rejects $H_0$ when in fact $\mu = 40$ dollars. (b) Find the probability of a Type II error when $\mu = 44.9$ dollars. This is the probability that your test accepts $H_0$ when in fact $\mu = 44.9$. (c) Find the probability of a Type II error when $\mu = 49.9$. (d) The distribution of sales is not normal, because many customers buy nothing. Why is it nonetheless reasonable in this circumstance to assume that the mean will be approximately normal? The sample size is very small. The sample size is very large. The mean is always normal. None of the above are true.

          You want to see if a redesign of the cover of a mail-order catalog will increase sales. A very large number of customers will receive the original catalog, and a random sample of customers will receive the one with the new cover. For planning purposes, you are willing to assume that the sales from the new catalog will be approximately normal with $\sigma = 65$ dollars and that the mean for the original catalog will be $\mu = 40$ dollars. You decide to use a sample size of $n = 990$. You wish to test the following hypotheses.
$H_0: \mu \le 40$
$H_a: \mu > 40$
You decide to reject $H_0$ if $\bar{x} > 43.08$ and to accept $H_0$ otherwise.
(a) Find the probability of a Type I error, that is, the probability that your test rejects $H_0$ when in fact $\mu = 40$ dollars.
(b) Find the probability of a Type II error when $\mu = 44.9$ dollars. This is the probability that your test accepts $H_0$ when in fact $\mu = 44.9$.
(c) Find the probability of a Type II error when $\mu = 49.9$.
(d) The distribution of sales is not normal, because many customers buy nothing. Why is it nonetheless reasonable in this circumstance to assume that the mean will be approximately normal?
The sample size is very small.
The sample size is very large.
The mean is always normal.
None of the above are true.

You want to see if a redesign of the cover of a mail-order catalog will increase sales. A very large number of customers will receive the original catalog, and a random sample of customers will receive the one with the new cover. For planning purposes, you are willing to assume that the sales from the new catalog will be approximately normal with σ = 65 dollars and that the mean for the original catalog will be μ = 40 dollars. You decide to use a sample size of n = 990. You wish to test the following hypotheses.
H0: μ≤ 40
Ha: μ > 40
You decide to reject H0 if x̅ > 43.08 and to accept H0 otherwise.
(a) Find the probability of a Type I error, that is, the probability that your test rejects H0 when in fact μ = 40 dollars.
(b) Find the probability of a Type II error when μ = 44.9 dollars. This is the probability that your test accepts H0 when in fact μ = 44.9.
(c) Find the probability of a Type II error when μ = 49.9.
(d) The distribution of sales is not normal, because many customers buy nothing. Why is it nonetheless reasonable in this circumstance to assume that the mean will be approximately normal?
The sample size is very small.
The sample size is very large.
The mean is always normal.
None of the above are true.

Added by Mark T.

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