Question

# 5) • z-value should be expressed to three decimal places: e.g., z=1.645. • Before you start...

5)

• z-value should be expressed to three decimal places: e.g., z=1.645.

• Before you start each question, you should determine if you are dealing with the population mean or proportion since what you do depends on the type of parameter at hand.

• When handling proportion problems, write/use/plug proportions as decimals (e.g., not 54%, but .54).

• In the beginning, state clearly if you do a 1-tailed or 2-tailed test. State clearly your H0 and H1 using correct Greek letters, mathematical symbols, and subscripts. Do not state your hypotheses in prose form. Express numbers to the three decimal places.

• Use Z test, not t test. Don’t use p-value. Articulate if you reject H0. Justify why you reject or FTR H0.

• After articulating your statistical decision, state clearly your research conclusion in prose form: do NOT use Greek letters, mathematical symbols, or subscripts for the research conclusion. Show all the relevant quantities (e.g., sample statistics, SE, Z-statistic, to name a few) and label them properly: don’t omit anything.

• At the end of your answer, compute and interpret the corresponding CI.

• Write the lower and upper bounds in the CI for μ to the two decimal places: e.g., [11.33, 15.55].

• Write the lower and upper bounds in the CI for π (or P) to the four decimal places: .e.g., [.3012, .3875].

5. A filling machine in a brewery is designed to fill bottles with 355 ml of hard cider. In practice, however, volumes vary slightly from bottle to bottle. In a sample of 49 bottles, the mean volume of cider is found to be 353.3 ml, with a standard deviation of 3.5 ml. At the 98% confidence level, what is your conclusion?

We will use one sample z test as it is mentioned in the instructions that use z test not t test.

Null hypothesis Ho : u = 355

Alternate hypothesis H1 : u not equal to 355

This is two tailed test

Alpha = 0.02 (1-0.98)

As the test is two tailed we will first divide the alpha into two equal parts

0.02/2 = 0.01

From z table, P(z<-2.33) = p(z>2.33) = 0.01

That is critical values are -2.33 and 2.33 respectively

And rejection region is

Reject null hypothesis Ho if test statistics is greater than 2.33 or less than -2.33

Test statistics z = (sample mean - claimed mean)/(s.d/√n)

Z = (353.3-355)/3.5/√49) = -3.4

As -3.4 is < -2.33

We reject null hypothesis Ho

Confidence interval.

Critical value z = 2.33

Margin of error (MOE) = z*s.d/√n

MOE = 2.33*3.5/√49 = 1.165

Confidence interval is given by

[Mean - MOE, Mean + MOE)

Mean = 353.3

[352.14, 354.47]

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