Scenario: A certain brand of apple juice is supposed to have 64 ounces of juice.

Question

Scenario: A certain brand of apple juice is supposed to have 64 ounces of juice. Because the filling machine is not precise, the exact amount of juice varies from bottle to bottle with a standard deviation = 0.06. However, the average amount of juice should be 64 ounces. The quality control manager occasionally needs to randomly select bottles and measure their contents to see if the machine is under- or over-filling. You are going to job-shadow her for the next few questions. Be sure to provide a complete statistical justification for each of your answers. a. (3 points) When you interview the quality control manager, she tells you that historically the probability of randomly selecting a bottle containing less than 63 ounces of juice is approximately 0%. What kind of probability would this be (classical, empirical, or subjective), and why? b. (3 points) She picks a bottle of apple juice off the line, measures the contents, and tells you that the juice content is about 1.5 standard deviations below average. Approximately how much juice is in the bottle? c. (3 points) As part of her routine, she now samples 22 bottles of juice. The results are as follows: 63.97 63.87 64.03 63.95 64.02 63.95 64.01 63.9 64 64.01 63.92 63.94 63.9 64.05 63.9 64.01 63.91 63.92 63.93 63.97 63.93 63.98 The statistical analysis tools that she will use with this sample require that the sampling distribution of the sample mean be normal. (We will learn about these tools in the next unit.) Why can she be confident that the normality requirement has been met? Provide all necessary supporting information/explanation. Hint: review the sections regarding sampling distributions. d. (2 points) Before finishing the last few problems, we should organize all of the information we have so far. Give the values for the following: (the first one has been done for you, and you do not need to show any “work” for any of these. Sample Statistics = 63.958 (found using Minitab) s = _______ n = _______ Population Parameters µ = ______ = ______ e. (3 points) Find the mean and standard deviation/standard error of the sampling distribution of . f. (5 points) The sample mean for the 22 bottles is = 63.958 (found using Minitab’s Descriptive Statistics). What is the probability that a random sample of 22 bottles has a mean content of 63.958 or less? Include a sketch of the sampling distribution with the area of interest shaded. g. (3 points) Is it unusual to observe a sample mean of 63.958 or less when the average content is supposed to be 64? (In other words, if the machine is working correctly and filling to an overall average of 64 ounces, is 63.958 the kind of sample average we would expect to see?) As a result, should the manager be concerned that the machine needs to be recalibrated?

Explanation / Answer

a) The probability value provided by the control manager is subjective , because it is generated as per her belief or past experience

b) Level = Mean - 1.5*SD

Mean = 64

Sd = 0.06

Level = 64 -1.5*0.06

Level = 64 -0.09

Level = 63.91

.

c) The first and foremost requirement is that the sample must be drawn randomly

this supports the assumption that the condition of normality is satisfied

we can also note that each observation is indpendent of the other observation.

Thus the assumptions of normality are satsified

d) I have made use of excel to find the sample statistics:

Enter the data in excel

Then type the commands

=AVERAGE(A1:A20)

this gives us the average 63.955

=STDEV(A1:A20)

this gives us the value of s = 0.050835

the value of n = 20

we got the population mean = 64

population standard deviation = 0.06

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