Question

To study the effect the ecological impact of malaria, researchers in California measured the effect of malaria on what distance an animal could run in 2 minutes. They used a sample of a local lizard species, Sceloporis occidentalis, collected in the field. They selected 15 lizards from those found to be infected with the malarial parasite Plasmodium mexicanum and 15 lizards found not infected.

The distance each lizard could run in a time limit of 2 minutes was recorded in a controlled environment. The data (below) is also in an Excel file, “Malaria effect”

a) Does this design use paired data or independent samples? Explain.

b) Compute a 95% confidence interval for the difference in distance ran between the two malarial infections of the lizards. Show all of your working.

c) Based on this confidence interval, can you conclude there is a difference in mean distance ran between the infected and uninfected lizards?

Infected	16.4	29.4	37.1	23.0	24.1	24.5	16.4	29.1	36.7	28.7	30.2	21.8	37.1	20.3	28.3
Uninfected	22.2	34.8	42.1	32.9	26.4	30.6	32.9	37.5	18.4	27.5	45.5	34.0	45.5	24.5	28.7

d) Conduct a test of significance (at a significance level of 5%) on the difference between infected and uninfected lizards to decide whether the uninfected lizards can run a further distance on average. Follow all steps clearly and write a clear conclusion.

e) Do you have the same conclusion from the confidence interval calculation in c) as that concluded by the test done in d) above? Explain why or why not.

Answer 1

a) DESIGN USES PAIRED OR INDEPENDENT DATA:

The unpaired or independent data arise from separate individuals. That is, the data is collected for two separate set of observations/samples. Whereas paired data arises from the same individual at different points in time. That is, the data is collected for sample set of sample/ observations at different points of time.

Thus in the given study, one set of 15 lizards (sample) are selected from those found to be infected with the malarial parasite Plasmodium mexicanum and not the same but different set of 15 lizards sample which is not infected are selected to be the other sample. And they are tested to measure the effect of malaria on what distance an animal could run in 2 minutes. Thus the design use independent samples.

b) 95% CONFIDENCE INTERVAL FOR THE DIFFERENCE IN MEAN DISTANCE BETWEEN TWO MALARIAL INFECTIONS OF LIZARDS:

The 95% confidence interval for difference in two means with unequal variances is given by,

Thus

Sample size of Infected lizards = 15

Sample size of Uninfected lizards = 15

Sample mean of Infected lizards ​

​

Sample mean of Uninfected​ lizards

Sample Standard deviation of Infected lizards :

= 15​ and ​

16.4	-10.47	109.62
29.4	2.53	6.40
37.1	10.23	104.65
23.0	-3.87	14.98
24.1	-2.77	7.67
24.5	-2.37	5.62
16.4	-10.47	109.62
29.1	2.23	4.97
36.7	9.83	96.63
28.7	1.83	3.35
30.2	3.33	11.09
21.8	-5.07	25.70
37.1	10.23	104.65
20.3	-6.57	43.16
28.3	1.43	2.04

Sample Standard deviation of Infected lizards

  

  

Sample Standard deviation of Uninfected lizards :

= 15 and

22.2	-10.03	100.60
34.8	2.57	6.60
42.1	9.87	97.42
32.9	0.67	0.45
26.4	-5.83	33.99
30.6	-1.63	2.66
32.9	0.67	0.45
37.5	5.27	27.77
18.4	-13.83	191.27
27.5	-4.73	22.37
45.5	13.27	176.09
34.0	1.77	3.13
45.5	13.27	176.09
24.5	-7.73	59.75
28.7	-3.53	12.46

Sample Standard deviation of Uninfected lizards

  

  

The degrees of freedom for two tailed t critical value ​ is given by,

Thus two tailed t critical value ​ at 5% level of significance and 27 degrees of freedom is 2.052.

Thus the 95% confidence interval for difference in two means with unequal variances is given by,

Thus the 95% confidence interval for difference in distance ran between two malarial infections is given by

INFERENCE:

Our best estimate of the difference, the point estimate, is -5.36 units. The standard error of the difference is 2.73 units and the margin of error is 5.60 units. We are 95% confident that the difference in mean distance between infected and uninfected lizards is between -10.96 and 0.24 units. Based on this interval, we also conclude that there is no statistically significant difference in mean distance between infected and uninfected lizards, because the 95% confidence interval includes the null value, zero. Again, the confidence interval is a range of likely values for the difference in means. Since the interval contains zero (no difference), we do not have sufficient evidence to conclude that there is a difference.

c)TWO INDEPENDENT SAMPLE T-TEST:

The independent sample t-test is a member of the t-test family, which consists of tests that compare mean value(s) of continuous-level(interval or ratio data), normally distributed data. ​Since the problem is to test on the difference between infected and unifected lizards we make use of two independent sample t test.

HYPOTHESIS:

​: There is no significant mean difference in distance ran between infected and uninfected lizards.

​: There is significant mean difference in distance ran between infected and uninfected lizards.

LEVEL OF SIGNIFICANCE :

TEST STATISTIC:

CALCULATION:

​   

​​CRITICAL VALUE:

The t critical value at 5% level of significance with 27 degrees of freedom is 2.052.

INFERENCE:

Since the calculated t value (-1.96) is less than t critical value (2.052), we fail to reject null hypothesis and conclude that there is no significant difference in distance ran between uninfected and infected lizards.

e) CONCLUSION FROM TEST AND CONFIDENCE INTERVAL:

Our best estimate of the difference, the point estimate, is -5.36 units. The standard error of the difference is 2.73 units and the margin of error is 5.60 units. We are 95% confident that the difference in mean distance between infected and uninfected lizards is between -10.96 and 0.24 units. Based on this interval, we also conclude that there is no statistically significant difference in mean distance between infected and uninfected lizards, because the 95% confidence interval includes the null value, zero. Again, the confidence interval is a range of likely values for the difference in means. Since the interval contains zero (no difference), we do not have sufficient evidence to conclude that there is a difference.

Even from the two sample independent t test we conclude that there is no statistically significant difference in mean distance between infected and uninfected lizards​. Thus we have same conclusion from confidence interval and t test.