State the null an alternative hypothesis in the following situations by defining the parameters used. Also, state any assumptions that you need to make to conduct the test: (a) The postal service wishes to prove that the mean delivery time for packages is less than 5 days. (b) A financial institution believes that it has an average loan processing time of less than 10 days. (c) A marketing firm believes that the average contract for a customer exceeds $50,000. (d) A Web-order company wished to test if it has improved its efficiency of operations by reducing its average response time. (e) A manufacturer of consumer durables believes that over 70% of its customers are satisfied with the product.

a) Let X the random variable who represent the mean delivery time for packages. They want to test if this time is lss than 5 days (alternative hypothesis) so we have:

Null hypothesis:

Alternative hypothesis:

b) Let X the random variable who represent the loan processing time. They want to test if this time is less than 10 days (alternative hypothesis) so we have:

Null hypothesis:

Alternative hypothesis:

c) Let X the random variable who represent the amoutn of money for the contract of a customer. They want to test if this amount exceeds 50000$ (alternative hypothesis) so we have:

Null hypothesis:

Alternative hypothesis:

d) Let's assume that the average response time is and X represent the variable the response time, so we want to check this:

Null hypothesis:

Alternative hypothesis:

e)For this case we are interested on the proportion of customers satisfied with a product and we want to test if this proportion is higher than 0.7 or 70% so the system of hypothesis should be:

Null hypothesis:

Alternative hypothesis:

Step-by-step explanation:

Part a

Let X the random variable who represent the mean delivery time for packages. They want to test if this time is less than 5 days (alternative hypothesis) so we have:

Null hypothesis:

Alternative hypothesis:

Part b

Let X the random variable who represent the loan processing time. They want to test if this time is less than 10 days (alternative hypothesis) so we have:

Null hypothesis:

Alternative hypothesis:

Part c

Let X the random variable who represent the amoutn of money for the contract of a customer. They want to test if this amount exceeds 50000$ (alternative hypothesis) so we have:

Null hypothesis:

Alternative hypothesis:

Part d

Let's assume that the average response time is and X represent the variable the response time, so we want to check this:

Null hypothesis:

Alternative hypothesis:

Part e

For this case we are interested on the proportion of customers satisfied with a product and we want to test if this proportion is higher than 0.7 or 70% so the system of hypothesis should be:

Null hypothesis:

Alternative hypothesis: