In order to assess the risk of impaired neurocognitive performance due to repeat
ID: 3183094 • Letter: I
Question
In order to assess the risk of impaired neurocognitive performance due to repeated concussive events, investigators interviewed 94 randomly selected soccer athletes, 93 randomly selected non-soccer athletes, and 47 randomly selected non-athletes. This last group served as the control group. The main interest for the investigative team was whether there was compelling evidence for the existence of chronic neuropsychological dysfunction in the population of collegiate soccer players. Of interest here is whether or not the rates for concussive events are distributed evenly for soccer athletes, non-soccer athletes, and the controls. Table 1 provides cross-tabulated results for this study. Let = 0.1.
Note: To calculate the chi squared bits, plug in the observed counts as you see them and plug in the ROUNDED (to two decimal places) expected counts into the chi squared bits equation.
D) The critical value is: ?
E) The test statistic is: ?
G) Suppose that the researchers now want to find a 90% confidence interval for the true difference in proportion suffering from 3 or more concussions for soccer athletes vs. the proportion suffering from 3 or more concussions for non-soccer athletes. Find each of the following:
Note: Calculate as soccer - non-soccer
a) The margin of error: ?
b) The 90% confidence interval: ? to ?
H) Suppose that the investigators now want to perform a hypothesis test (still at the = 0.1 level) to see if the proportion of non-athletes that do not experience any concussions is greater than the proportion of soccer players that do not experience any concussions.
Note: If the test statistic is not on your z-table (e.g., it is too large), I would recommend using your TI-83/84 function 'normalcdf' to find the pvalue. You can find instructions on how to find the pvalue on your calculator on RamCT or you can send me an email or come into office hours.
a) The proper null/alternative pair for this setting is:
* Ho: soccer - non-athlete; Ha: soccer - non-athlete 0
* Ho: soccer - non-athlete 0; Ha: soccer - non-athlete < 0
* Ho: soccer - non-athlete 0; Ha: soccer - non-athlete > 0
b) The test statistic is: ?
c) The p-value is: ?
d) The proper interpretation is:
* Reject Ho and conclude that a greater proportion of non-athletes than soccer players do not experience concussions.
* FTR Ho and do not conclude that a greater proportion of non-athletes than soccer players do not experience concussions.
* FTR Ho and do not conclude that a greater proportion of soccer players than non-athletes do not experience concussions.
* Reject Ho and conclude that a greater proportion of soccer players than non-athletes do not experience concussions.
Table 1: Cross-tab results for number of concussions by athlete type Number of Concussions 0 1 2 3 or more Type 10 Soccer Exp11 63.47 Exp14. x211 5.37 12 10.72 73 Non- Exp21 62.79 7.95 Exp24. Soccer X 23 .0003 21 1.66 40 Non- Exp31 31.74 9.04 Exp34. Athlete 32 2.81 X 31 2.15 45 158 Total 20 11 Total 94 234Explanation / Answer
Answer:
D) The critical value is: 10.645
E) The test statistic is: 31.23
G) Suppose that the researchers now want to find a 90% confidence interval for the true difference in proportion suffering from 3 or more concussions for soccer athletes vs. the proportion suffering from 3 or more concussions for non-soccer athletes. Find each of the following:
Note: Calculate as soccer - non-soccer
a) The margin of error: 0.0538
b) The 90% confidence interval: -0.0116 to 0.096
p1
p2
pc
0.0745
0.0323
0.0535
p (as decimal)
7/94
3/93
10/187
p (as fraction)
7.
3.
10.
X
94
93
187
n
0.0422
difference
0.0329
std. error
-0.0116
confidence interval 90.% lower
0.096
confidence interval 90.% upper
0.0538
margin of error
H) Suppose that the investigators now want to perform a hypothesis test (still at the = 0.1 level) to see if the proportion of non-athletes that do not experience any concussions is greater than the proportion of soccer players that do not experience any concussions.
Note: If the test statistic is not on your z-table (e.g., it is too large), I would recommend using your TI-83/84 function 'normalcdf' to find the pvalue. You can find instructions on how to find the pvalue on your calculator on RamCT or you can send me an email or come into office hours.
a) The proper null/alternative pair for this setting is:
* Ho: soccer - non-athlete; Ha: soccer - non-athlete 0
Answer* Ho: soccer - non-athlete 0; Ha: soccer - non-athlete < 0
* Ho: soccer - non-athlete 0; Ha: soccer - non-athlete > 0
* Ho: soccer - non-athlete 0; Ha: soccer - non-athlete < 0
* Ho: soccer - non-athlete 0; Ha: soccer - non-athlete > 0
* Ho: soccer - non-athlete = 0; Ha: soccer - non-athlete 0
b) The test statistic is: -4.26
c) The p-value is: 0.0000
d) The proper interpretation is:
Answer: * Reject Ho and conclude that a greater proportion of non-athletes than soccer players do not experience concussions.
* FTR Ho and do not conclude that a greater proportion of non-athletes than soccer players do not experience concussions.
* FTR Ho and do not conclude that a greater proportion of soccer players than non-athletes do not experience concussions.
* Reject Ho and conclude that a greater proportion of soccer players than non-athletes do not experience concussions.
p1
p2
pc
0.4787
0.8511
0.6028
p (as decimal)
45/94
40/47
85/141
p (as fraction)
45.
40.
85.
X
94
47
141
n
-0.3723
difference
0.
hypothesized difference
0.0874
std. error
-4.26
z
1.02E-05
p-value (one-tailed, lower)
p1
p2
pc
0.0745
0.0323
0.0535
p (as decimal)
7/94
3/93
10/187
p (as fraction)
7.
3.
10.
X
94
93
187
n
0.0422
difference
0.0329
std. error
-0.0116
confidence interval 90.% lower
0.096
confidence interval 90.% upper
0.0538
margin of error
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