Question

A syrup company sells pure maple syrup in 16-ounce bottles. As part of the company’s quality control program, random sampling is done weekly. If too little syrup is put in each bottle, customers will be dissatisfied, and if too much syrup is put in each bottle, the company will lose money. In either case, the equipment would be turned off to investigate the problem further. Testing is conducted using a 5% significance level. (a) State the appropriate hypotheses. (b) What assumptions are required to perform the test stated in part (a)? (c) How would a Type I error affect the company? (d) How would a Type II error affect the company? (e) The week’s sample of 20 bottles had a mean of 15.8 ounces and a standard deviation of 0.7 ounces. Perform the test, and report the associated p-value. (f) Based on the data and res ults of the test, what would be your recommendation?

Answer 1

(a) The Hypothesis:

H0: = 16 ounces : The amount of maple syrup being filled into each bottle is equal to 16 ounces.

Ha: 16 ounces : The amount of maple syrup being filled into each bottle is different from 16 ounces.

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(b) The Assumptions

(i) The sampling does is Random.

(ii) Each sample is independent of the other.

(ii) The sample come from a population that is normal or approximately normal.

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The Errors:

(c) Type I error: The error that occurs when we reject a true null hypothesis. In this case it would mean that the equipment is filling the correct amount of 16 ounces, but we reject that and conclude that the amount being filled is different from 16 ounces.

The effect of this error would be that production lines would be closed when actually it is not required to be closed, causing losses to the company in terms of sales and checking the equipment for errors where there aren't any.

(d) Type II error: The error that occurs when we fail to reject a false null hypothesis. In this case it would mean that the equipment is not filling the correct amount of 16 ounces, but we reject that and conclude that the amount being filled is equal to 16 ounces.

The effect of this error would be that bottles will not be filled with the correct amount of 16 ounces and the process would continue when actually it is required to be shut down.

If the bottles are being under filled (< 16 ounces)This would lead to customer dissatisfaction and which may the lead to legal action, or create a negative brand name for the company if it is found that they are not living up to their claim.

If the bottles are being over filled, then the company will have losses as they are charging for 16 ounces only, when actually they are giving more than 16 ounces.

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(e) The Test Statistic: Since the population standard deviation is unknown, we use the students t test.

The test statistic is given by the equation:

t = \frac{\bar{x}-\mu }{\frac{s}{\sqrt{n}}} = \frac{15.8-16}{\frac{0.7}{\sqrt{20}}} = -1.28

t observed = -1.28

The p Value:    The p value for t = -1.28, for degrees of freedom (df) = n-1 = 19, is; p value = 0.2160

The Critical Value:   The critical value (2 Tail) at = 0.05, for df = 19, tcritical= +2.09 and -2.09

The Decision Rule:   If tobserved is > tcritical or If tobserved is < -tcritical, Then reject H0.

Also if P value is < , Then Reject H0.

The Decision:   Since tobserved (-1.28) falls in between +2.09 and -2.09, We Fail to Reject H0.

Also since P value (0.2160) is > (0.05) , We Fail to Reject H0.

The Conclusion: There isn'rt sufficient evidence at the 95% significance level to conclude that the amount of maple syrup being filled into each bottle is different from 16 ounces.

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(f) My recommendation would be that there is no further investigation for the present and to continue with the equipment as it is and not to shut down the process.

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