Suppose a company manufactures ropes that are known to, 011 average, break when

Question

Suppose a company manufactures ropes that are known to, 011 average, break when supporting 75kg. An engineer claims that her ropes are superior (have a higher break point), but they also cost substantially more to manufacture. In the context of a hypothesis test, we would make the following assumption about the new rope H_0 : mu = 75 and look for evidence of her claim : mu > 75. Let's assume sigma is known to be 9 for the new ropes. We can think of that hypothesis test as a proxy for deciding whether to start manufacturing the new ropes. Consider the alternative value mu = 76, which in the context of the problem would presumably not be a practically significant departure from H_0 - that is, given the extra cost of the ropes, the relatively small increase in average durability is not worth it. Also, from an individual rope standpoint (with a standard deviation of 9kg) the durability increase would not be noticeable to a consumer. (a) For an a level .01 test, compute beta at this alternative for sample sizes n = 100, 900, and 2500. (b) If the observed mean of is x = 76, what can you say about the resulting p-value when n = 2500? Is the data statistically significant at any of the standard values of alpha? (c) Would you want to use a sample size of 2500 along with a level alpha = .01 test (disregarding the cost of such an experiment)? Explain.

Explanation / Answer

Solution:

a. Using Z-tables, the respective Z-score with p < 0.01 is 2.33

The formula for is 1 – N(-2.33 + n/9)

For n = 100,

1 - N (-2.33 +  100/9)

1 - N (-2.33 + 10/9)

1 - N (-1.2189)

0.8886

For n = 900,

1 - N (-2.33 +  900/9)

1 - N (-2.33 + 30/9)

1 - N (1.00)

0.1579

For n = 2500,

1 - N (-2.33 +  2500/9)

1 - N (-2.33 + 50/9)

1 - N (3.23)

0.0006

b. The respective Z-score with x-bar = 76 is

Z = (x-bar - µ)/ (/n)

Z = (76 - 75)/ (9/2500)

Z = 5.56

Which is “off the z table,” so p-value < .0002; this value of z is quite statistically significant.

c. No. Even when the higher break point from Ho is insignificant from a practical point of view, a statistically significant result is highly likely to appear; the test is too likely to detect lower break point from Ho.

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