Chapter 9 Extra Practice

1. Publishing scientific papers online is fast, and the papers can be long. Publishing in a paper journal means that the paper will live forever in libraries. The British Medical Journal combines the two: it prints short and readable versions, with longer versions available online. Is this OK with authors? The journal asked a random sample of 104 of its recent authors several questions. One question was “Should the journal continue using this system?” In the sample, 72 said “Yes.” a) Do the data give good evidence that more than two-thirds (67%) of authors support continuing this system? Carry out an appropriate test to help answer this question.

a) State: We want to perform a test at the α = 0 a) State: We want to perform a test at the α = 0.05significance level of H0: p = 0.67 Ha: p > 0.67, where p = the true proportion of authors who support the system of only publishing longer papers online. Plan: The procedure is a one-sample z-test for a proportion. Random: The journal took a random sample of 104 recent authors. 10%: Since we are sampling without replacement, the number of authors of articles in the British Medical Journal must be at least 1040. It is reasonable to assume that there are at least 1040 recent authors of articles in the British Medical Journal.

Large Counts/Normal: Assuming H0: p = 0. 67 is true, np = 104(0 Large Counts/Normal: Assuming H0: p = 0.67 is true, np = 104(0.67) = 69.7 ≥ 10, and n(1 – p) = 104(0.33) = 34.3 ≥ 10. Because both of these values are at least 10, we should be safe doing Normal calculations. Do: The sample proportion of authors who support the system of only publishing longer papers online is 0.6923 P(z > 0.48) = normalcdf(lower: 0.48, upper: ∞, µ: 0, σ: 1) = 0.3143 Conclude: Fail to reject H0. Since the P-value, 0.3143, is greater than α = 0.05, there is not enough evidence to conclude that the proportion of recent authors who support the system is greater than 67%.

b) Interpret the P-value from your test in the context of the problem b) Interpret the P-value from your test in the context of the problem. b) Assuming the proportion of authors who support the system is 0.67 is true, then the likelihood of getting a sample of size 104 with a sample proportion value of 0.69 or greater is roughly 0.3143.

2. “Red tide” is a bloom of poison-producing algae—a few different species of a class of plankton called dinoflagellates. When weather and water condition cause these blooms, shellfish such as clams living in the area develop dangerous levels of a paralysis-inducing toxin. In Massachusetts, the Division of Marine Fisheries (DMF) monitors levels of the toxin in shellfish by regular sampling of shellfish along the coastline. If the mean level of toxin in clams exceeds 800 mg (micrograms) of toxin per kg of clam meat in any area at a 5% level of significance, clam harvesting is banned there until the bloom is over and levels of toxin in clams subside. During a bloom, the distribution of toxin levels in clams on a single mudflat is distinctly non-Normal. a) Define the parameter of interest and state appropriate hypotheses for the DMF to test.

a) H0: µ = 800 Ha: µ > 800, where µ = the mean concentration of Red Tide toxins in clams, in mg/kg. b) Because of budget constraints and the large number of coastal areas that must be tested, the DMF would like to sample no more than 10 clams from any single area. Explain why this sample size may lead to problems in carrying out the significance test from (a). b) This sample size is too small for a population that is known to be nonnormal.

c) Describe a Type I and a Type II error in this situation and the consequences of each. c) Type I error: Concluding that the mean level of toxin is above 800 mg/kg when it is equal to 800 mg/kg. Consequence: The DMF would close the area to clam harvesting, which would have a negative economic impact on anyone who depends on the clam business, even though the clams are safe to eat. Type II error: Concluding that the mean level of toxin is 800 mg/kg when it above 800 mg/kg. Consequence: This could cause anyone who eats clams from this area to become sick or even die.