In: Statistics and Probability
For each of the following, (a) state the null hypothesis and (b) state the alternate hypothesis, (c) state the type of test that you will use to test the hypothesis and the level of significance, (d) state the critical value , (e) calculate the z-statistic or the t-statistic, (f) state whether we shall reject or fail to reject the null hypothesis, (g) state the relation between the p-value and α with an appropriate inequality, and (h) conclude the hypothesis test with an appropriate summarizing statement addressing the alternate hypothesis.
1) The null hypothesis is Ho : mu >=115
Ha : mu < 115
Level of significance is 5% and test is one sided.
Test statistic is t = n1/2 *(Xbar - mu) /s with degree of freedom n-1
X bar = sample mean = 105 (calculated)
n = sample size = 6
s = sample standard deviation = 11.26 ( calculated with df =5)
Calculated value of t = -10* 2.449 / 11.26 = -2.17
The mod of this value is 2.17
The pvalue is 0.041072
The Critical value of t is calculated with degree of freedom =5 and alpha =0.05 This is found to be 2.015 (from t table)
As mod calculated t > t tabulated we reject the null hypothesis
Also as p < level of significance (0.041072 <5) we reject the null hypothesis
Hence we conclude that there is a significant slowing in the heart beat of the cocker spaniel.
2) Here Ho : P =0.238
Ha : p is not equal to 0.238
Level of significance is 5% (two sided)
Test statistic is Z = p-P /(PQ/n)1/2
where p = sample proportion = 178/610 = 0.291
P = population proportion = 0.238
Q= 1-P =0.762
n= 610
Putting this we get Z =3.08
Critical value of Z is 1.96 (corresponding to right tail of 2.5%)
p value of getting a Z score of 3.06 is 0.00207
Hence as Zcalc > Z tab we reject the null hypothesis
Also as p < 2.5% we reject the null hypothesis
We thus conclude that the proportion is significantly different form 23.8%