Answer Key

Hurricane Strengths

1. α=0.10.
2. H₀: “The distribution of hurricanes into the strength categories is the same for both time periods” versus H_A: “The distribution of hurricanes into the strength categories is NOT the same for both time periods”
3. A chi-square test is required because (i) a categorical response variable with three levels (strength category) from (ii) two groups or populations (time periods) was recorded.
4. This is an observational study without obvious randomization.
5. The test statistic computed below should reasonably follow a chi-square distribution because all cells in the expected table (Table 1) have values greater than five.

Table 1: Expected frequency table for distribution of hurricanes into the strength and time period categories.

          Cat 1&2 Cat 3 Cat 4&5
1901-1950    51.4  25.4     8.1
1951-2000    43.6  21.6     6.9

6. The table of observed frequencies were given as in Table 2.

Table 2: Observed frequency table for distribution of hurricanes into the strength and time period categories.

          Cat 1&2 Cat 3 Cat 4&5 Sum
1901-1950      51    26       8  85
1951-2000      44    21       7  72
Sum            95    47      15 157

7. The χ² test statistic is \(\frac{(51-51.4)^{2}}{51.4}\) + \(\frac{(26-25.4)^{2}}{25.4}\) +\(\frac{(8-8.1)^{2}}{8.1}\) +\(\frac{(44-43.6)^{2}}{43.6}\) +\(\frac{(21-21.6)^{2}}{21.6}\) +\(\frac{(7-6.9)^{2}}{6.9}\) = 0.0036+0.0121+0.0018+0.0043+0.0142+0.0021 = 0.0382 with 2 df.
8. The p-value is 0.9811.
9. The H₀ is not rejected because the p-value>α.
10. It appears that the distribution of hurricanes into the strength categories is the same for both time periods (Table 3).

Table 3: Row percentage table for the observations of hurricane strengths by time period.

          Cat 1&2 Cat 3 Cat 4&5 Sum
1901-1950    60.0  30.6     9.4 100
1951-2000    61.1  29.2     9.7 100

11. Not required for a chi-square test.

R Appendix.

library(NCStats)
distrib(0.0382,distrib="chisq",df=2,lower.tail=FALSE)

Response to Hello

1. α = 0.05.
2. The H₀: “The distribution of groups into whether they responded or not to the ‘Hello’ is the same for all three group sizes” versus H₀: “The distribution of groups into whether they responded or not to the ‘Hello’ is NOT the same for all three group sizes.”
3. A chi-square test is required because (i)a categorical variable with two levels (response or not) was measured on (ii) three groups/populations (group sizes).
4. This study is experimental in that the the researchers intervened with the subjects (i.e., said “Hello” to them) but it is observational in the sense that the “groups” (number of peple together) were not created by the researcher. It is clear that the “groups” were not randomly selected.
5. The test statistic below should follow a χ² distribution because the expected number in each cell is greater than five (Table 4).

Table 4: Expected frequencies of people that responded to ‘Hello’ and group size.

             Responded Did Not Respond
Individual        84.3            34.7
Two or Three      66.6            27.4
Four or more      19.1             7.9

6. The observed frequency table is in Table 5.

Table 5: Observed frequencies of people that responded to ‘Hello’ and group size.

             Responded Did Not Respond Sum
Individual          92              27 119
Two or Three        65              29  94
Four or more        13              14  27
Sum                170              70 240

7. The χ² test statistic is \(\frac{(92-84.3)^{2}}{84.3}\) + \(\frac{(27-34.7)^{2}}{34.7}\) +\(\frac{(65-66.6)^{2}}{66.6}\) +\(\frac{(29-27.4)^{2}}{27.4}\) +\(\frac{(13-19.1)^{2}}{19.1}\) +\(\frac{(14-7.9)^{2}}{7.9}\) = 0.703+ 1.709+ 0.038+ 0.093+ 1.948+ 4.710= 9.201 with (3-1)(2-1)=2 df.
8. The p-value is 0.0100.
9. The H₀ is rejected because the p-value < α.
10. There is a difference in the responsiveness among the three sizes of groups. It appears that the groups with 4,5,6 individuals responded less frequently than the individuals and groups of 2 and 3 (Table 6).

Table 6: The percentage of people that responded to ‘Hello’ by group size.

             Responded Did Not Respond Sum
Individual        77.3            22.7 100
Two or Three      69.1            30.9 100
Four or more      48.1            51.9 100

11. Not needed for a chi-square test.

R Appendix.

library(NCStats)
distrib(9.201,distrib="chisq",df=2,lower.tail=FALSE)

Turtle Excluder Devices

Table 7: Expected frequencies of trawl tows with at least one turtle mortality by two sizes of openings.

         mortality no mortality
original  11.04294     63.95706
new       12.95706     75.04294

Table 8: Observed frequencies of trawl tows with at least one turtle mortality by two sizes of openings.

         mortality no mortality
original        16           59
new              8           80

Table 9: The percentage of trawl tows with at least one turtle mortality by two sizes of openings.

         mortality no mortality Sum
original      21.3         78.7 100
new            9.1         90.9 100

> obs <- matrix(c(16,75-16,8,88-8),nrow=2,byrow=TRUE)
> rownames(obs) <- c("original","new")
> colnames(obs) <- c("mortality","no mortality")
> ( chi1 <- chisq.test(obs,correct=FALSE) )
> chi1$expected
> percTable(chi1$observed,margin=1)