Question

In the Northern Hemisphere, dolphins routinely swim in counterclockwise patterns while sleeping. The table below is the number of minutes spent swimming in a counterclockwise pattern for 8 Northern dolphins over a period of two hours.
77.7 84.8 79.4 84.0 99.6 93.6 89.4 97.2

a. Calculate the true 95% confidence interval for the average amount of time that a Northern dolphins spends swimming in a counterclockwise pattern in a two hour period.



Researchers wanted to see if Southern hemisphere dolphins spent the same or a different amount of time swimming in a clockwise pattern while sleeping. The number of minutes spent swimming in a counterclockwise pattern in a two hour time period for 11 Southern dolphins is below.
63.4 58.6 73.9 68.8 83.2 71.5 96.4 68.3 76.4 81.4 75

Determine if the two populations are the same or different from each other using the two sample t-test.   

b. Clearly state your null and alternate hypothesis.


c. Show your calculations to calculate the t-statistic




d. Do you reject, or fail to reject, your null hypothesis? Why?

Answer 1

a. Calculate the true 95% confidence interval for the average amount of time that a Northern dolphins spends swimming in a counterclockwise pattern in a two hour period.

The population Std deviation is unknown hence we will use t-distribution

(1 - ) confidence interval for population mean when population Std deviation is unknown is calculated as follows

Where

n = 8

Critical value = 2.3646   using t-dist tables or excel function 'tinv(0.05,7)' or also online

The stats are calculated as follows

Sr.	North
1	77.7	-10.5125	110.5127
2	84.8	-3.4125	11.64516
3	79.4	-8.8125	77.66016
4	84	-4.2125	17.74516
5	99.6	11.3875	129.6752
6	93.6	5.3875	29.02516
7	89.4	1.1875	1.410156
8	97.2	8.9875	80.77516
Total	705.7	0	458.4488

Substituting in our interval equation

=

Determine if the two populations are the same or different from each other using the two sample t-test.   

To check for difference of means we have to conduct a hypothesis test for difference of population means. Since we don't have the population standard deviation we will again use the t-dist. Also since there are tow separate sets of samples, we can say that it independent t-test. We are checking for the difference (irrespective if one is greater than the other) hence it is a 2-tailed test.

b. Clearly state your null and alternate hypothesis.

( Northern and Southern hemisphere dolphins spent the same amount of time swimming in a clockwise pattern while sleeping.)
VS

( Northern and Southern hemisphere dolphins spent different amount of time swimming in a clockwise pattern while sleeping.)

c. Show your calculations to calculate the t-statistic

Test Statistic :

Sr.	South
1	63.4	-10.8636	118.0186
2	58.6	-15.6636	245.3495
3	73.9	-0.36364	0.132231
4	68.8	-5.46364	29.85132
5	83.2	8.936364	79.8586
6	71.5	-2.76364	7.637686
7	96.4	22.13636	490.0186
8	68.3	-5.96364	35.56496
9	76.4	2.136364	4.56405
10	81.4	7.136364	50.92769
11	75	0.736364	0.542231
Total	816.9	0	1062.465

The statistics

Statistics	North (n)	South (s)
n	8	11
	88.2125	74.263
	8.09276	10.3076

Pooled variance=

=

= 89.4655

=

=

d. Do you reject, or fail to reject, your null hypothesis? Why?

The critical value at

= 2.1098

Since test stat > Critical value (3.1738 > 2.1098)

Decision: We reject the null hypothesis at 5% level of significance and conclude that Northern and Southern hemisphere dolphins spent significantly different amount of time swimming in a clockwise pattern while sleeping.