Question

Please use R studio, Thank you.

2. The probability of a student passing statistics is known to be 0.41; and the probability of a student passing chemistry is known to be 0.55. If the probability of passing both is known to be 0.35, calculate:

(a) the probability of passing at least one of statistics and chemistry

(b) the probability of a student passing chemistry, given that they passed statistics

(c) Are passing chemistry and statistics independent? Justify

(d) (harder) a group of 33 randomly selected students attend a special seminar on study skills. Of these 33, only 7 fail both. State a sensible null hypothesis, test it, and interpret.

Answer 1

Let

S=the event of passing statistics

C=the event of passing chemistry

P(S)=0.41 P(C)=0.55

Hence not independent.

d)Let p denotes the proportion of success in seminaron study skills.

p-value=0.9991 quite close to1, hence at any significancy level we are not able to reject the null hypothesis hence we can strongly conclude that the proportion of success is quite larger than 0.5, i.e. rate of success is higher than the rate of failure.

R CODE:

> #a)
> p_S=0.41;p_C=0.55;p_SC=0.35
> p_S+p_C-p_SC
[1] 0.61
> #b)
> p_SC/p_S
[1] 0.8536585
> #c)
> p_S*p_C
[1] 0.2255
> #d)
> prop.test(33-7, 33, p = 0.5,alternative = "less")

   1-sample proportions test with continuity correction

data: 33 - 7 out of 33, null probability 0.5
X-squared = 9.8182, df = 1, p-value = 0.9991
alternative hypothesis: true p is less than 0.5
95 percent confidence interval:
0.0000000 0.8919282
sample estimates:
p
0.7878788