Natural cork in wine bottles is subject to deterioration, and as a result wine i

Question

Natural cork in wine bottles is subject to deterioration, and as a result wine in such bottles may experience contamination. The article "Effects of Bottle Closure Type on Consumer Perceptions of Wine Quality" (Amer. J. of Etiology and Viticulture, 2007: 182-191) reported that, in a tasting of commercial chardonnays, 16 of 91 bottles were considered spoiled to some extent by cork-associated characteristics. Does this data provide strong evidence for concluding that more than 15% of all such bottles are contaminated in this way? Carry out a hypothesis test (clearly write down each of the 4 steps for the hypothesis test) to decide whether more than 15% of all such bottles are contaminated. Use significance level of 0.05 Verify the test you use is appropriate. For example, if you use a t-test or z-test, verify the corresponding conditions.

Explanation / Answer

Part A

Here, we have to use z test for population proportion.

The null and alternative hypothesis for this test is given as below:

Null hypothesis: H0: p = 0.15

Alternative hypothesis: Ha: p > 0.15

The test statistic formula is given as below:

Z = (P – p) / sqrt(pq/n)

Where, P = sample proportion, p = population proportion, q = 1 – p, n = sample size

For the given data we have

Alpha = 0.05

Number of items of interest = x = 16

Sample size = n = 91

Sample proportion = 16/91 = 0.175824

Standard error = sqrt(pq/n) = sqrt(0.15*0.85/91) = 0.0374

Z = (0.175824 – 0.15) / 0.0374

Z = 0.6899

P-value = 0.2451

P-value > Alpha value

So, we do not reject the null hypothesis

There is insufficient evidence that more than 15% of all such bottles are contaminated.

Part b

Here, z test is appropriate.

Because two conditions are satisfied as below:

np = 91*0.15 = 13.65 > 5

nq = 91*0.85 = 77.35 > 5

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