A coffee machine is adjusted to provide a population mean of 110ml of coffee per cup and a standard deviation of 5 ml. The volume of coffee per cup is assumed to have a normal distribution. The machine is checked periodically by sampling 12 cups of coffee. If the mean volume, Y, of those 12 cups in ml falls in the interval (110- 2 cy~) < y < (110 + 2 o~ ), no adjustment is made. Otherwise, the machine is adjusted.

a) If a 12-cup test gives a mean volume of 107.0 ml, what should be done?

b) What fraction of the total number of 12-cup tests would lead to an adjust-

ment being made, even if the machine had not changed from its original

correct setting?

c) How many cups should be sampled randomly so there is 99% confidence

that the mean volume of the sample will lie within +2 ml of 110ml when the machine is correctly adjusted?

a)

The mean volume (107) is less than the lower limit of the interval (110-2 = 108). Thus, there would be some measurements which were below the lower limit of the interval. So adjustments should be made.

b)

Standard error of sample mean = \sigma / \sqrt{n} = 5 / \sqrt{12} = 1.443376

P(adjustment is made) = P(y < 110 - 2) + P(y > 110 + 2)

= P(y < 108) + P(y > 112)

= P(z < (108 - 110)/1.443376) + P(z > (112 - 110)/1.443376)

= P(z < -1.39) + P(z > 1.39)

= 0.0823 + 0.0823

= 0.1646

c)

z value for 99% confidence interval is 2.58

Margin of error, E = 2

Number of cups, n = (z * \sigma / E)2

= (2.58 * 5 / 2)2

= 41.6 \approx 42

4.6875 pounds

step-by-step explanation:

when functions are first introduced, you will probably have some simplistic "functions" and relations to deal with, usually being just sets of points. these won't be terribly useful or interesting functions and relations, but your text wants you to get the idea of what the domain and range of a function are. small sets of points are generally the simplest sorts of relations, so your book starts with those.

step-by-step explanation:

state the domain and range of the following relation. is the relation a function?

{(2, –3), (4, 6), (3, –1), (6, 6), (2, 3)}

the above list of points, being a relationship between certain x's and certain y's, is a relation. the domain is all the x-values, and the range is all the y-values. to give the domain and the range, i just list the values without duplication:

domain: {2, 3, 4, 6}

range: {–3, –1, 3, 6}

(it is customary to list these values in numerical order, but it is not required. sets are called "unordered lists", so you can list the numbers in any order you feel like. just don't duplicate: technically, repetitions are okay in sets, but most instructors would count off for this.)

while the given set does indeed represent a relation (because x's and y's are being related to each other), the set they gave me contains two points with the same x-value: (2, –3) and (2, 3). since x = 2 gives me two possible destinations (that is, two possible y-values), then this relation is not a function.

note that all i had to do to check whether the relation was a function was to look for duplicate x-values. if you find any duplicate x-values, then the different y-values mean that you do not have a function. remember: for a relation to be a function, each x-value has to go to one, and only one, y-value.

state the domain and range of the following relation. is the relation a function?

{(–3, 5), (–2, 5), (–1, 5), (0, 5), (1, 5), (2, 5)}

i'll just list the x-values for the domain and the y-values for the range: