Seasonal Weevil Migration

 

a) To understand migratory patterns of the imported longhorned weevil, look at the data in Table 1. Were the imported longhorned weevils more likely to be moving between the red clover and the woods or between the red clover and the pasture?

b) Offer some possible explanations for your findings in Part a.

c) Using the data from Table 1, what would be your estimate of the proportion of the population of crawling imported longhorn weevils that are in transit between the woods and the red clover (as opposed to those in transit between the pasture and the red clover)?

d) Calculate an approximate 95 percent confidence interval for the proportion of crawling weevils in transit between the woods and red clover based on the data in Table 1. Comment on the results.

e) Specify the population we are making inference about in Part d.

 

• Understand that in a long sequence of independent repetitions of a random phenomenon, the relative frequency of an outcome approaches a fixed number; that number is the probability of the outcome.
• Understand that the goal of inference is to use the sample statistics to estimate the population parameters with confidence.
• Be able to make confidence intervals for proportions and means from a simple random sample.
• Be able to interpret confidence intervals in lay terms.

a) From Table 1 we see that there were a total of 109 imported longhorned weevils trapped in 1990, 98 of which (89.9 percent) were either RC-Woods or Woods-RC. In 1991 a total of 61 imported longhorned weevils were trapped, 46 of which (75.4 percent) were either RC-Woods or Woods-RC. Since most of the trappings involved weevils either coming from or going to the woods (as opposed to the pasture) it certainly seems that the weevils are more likely to be moving between the red clover and the woods than between the red clover and the pasture.

 

b) One possible explanation is that imported longhorned weevil adults live in the woods and come out of the woods to eat red clover and/or breed. Another possible explanation is that the weevils living in the pasture have plenty of food there and don't need to crawl to the red clover to eat.

 

c) An estimate for p would be

From Table 1 we have

d) An approximate 95 percent confidence interval for p is

where z_.025 is the upper 2.5 percent standard normal critical value. Thus the 95 percent confidence interval for p is (0.7985, 0.9055). The confidence interval is far from containing 0.5, so it indicates that there are more weevils in transit between the woods and red clover than between the pasture and red clover.

 

e) We are making inference about a very specific population of imported longhorned weevils—those that crawl in the studied area during periods of peak activity. Recall that only crawling insects are trapped and that the experimental results were limited to certain peak periods of crawling activity.

a) The researchers suggest that during initial periods of imported longhorned weevil activity, a greater number of imported longhorned weevils moved from the red clover towards the woods rather than from the woods to red clover, and later, the number of imported longhorned weevils moving from the woods towards the clover was greater than the number moving from the red clover towards the woods. Use the combined data in Table 2 for 1990 and 1991 to test their claim.

b) The dates for the initial and later periods differed in 1990 and 1991. This occurred because of weather differences in the two years, and suggests conducting tests of association between type of trap and time period (initial or later) trapped for each year separately. Conduct tests of association for both 1990 and 1991. Do your results support the same hypothesis as in Part a?

c) Should we conclude that the migratory pattern suggested by Culik and Weaver (1994) is indeed the case for the imported longhorned weevil?

 

• Understand the underlying idea of methods for assessing the statistical significance of an observed association between two categorical variables such as a chi-square test for association (in a two-way table).
• Be able to carry out the chi-square test including stating the appropriate null and alternative hypotheses.
  

a) We perform a chi-square test of independence using the data from Table 2. The null and alternative hypotheses are:

H₀: There is no association between type of trap and time period (initial or later).

H_a: There is an association between type of trap and time period.

The data in Table 2 give a value of χ² = 21.7, which when compared to a chi-square distribution with 1 df has a P-value < 0.0001. This indicates that we have very strong evidence against the null hypothesis. We conclude that there is strong evidence of an association between the type of trap that an imported longhorned weevil was caught in and the time period in which it was caught.

 

b) The null and alternative hypotheses are the same as in Part a, namely

H₀: There is no association between type of trap and time period.

H_a: There is an association between type of trap and time period.

For the 1990 data we find that χ² = 13.2 with 1 df, P-value = 0.0003.
For the 1991 data we find that χ² = 8.8 with 1 df, P-value = 0.003.

In each case there is strong evidence in favor of rejecting the null hypothesis. The results support the same conclusion as in Part a. That is, there is an association between type of trap and time period. The data suggests that this association is in the direction the researchers proposed. Namely, that during initial periods of activity, greater numbers of imported longhorned weevils move from the red clover to the woods than from the woods to red clover, and later, greater numbers of imported longhorned weevils move from the woods to the red clover.

 

c) Two years showing the same pattern is certainly suggestive of the migratory pattern being as the researchers suggest. One should, however, always consider other possible confounding factors that might explain the weevils' movement.