Question

Data from a recent year showed that 69​% of the tens of thousands of applicants to a certain program were accepted. A company that trains applicants claimed that 186 of the 250 students it trained that year were accepted. Assume these trainees were representative of the population of applicants. Has the company demonstrated a real improvement over the​ average? Complete parts a through c below.

​a) What are the​ hypotheses?

Which hypotheses below will test if the company demonstrated improvement over the​ average?

A.

Upper H 0H0​:

pequals=0.690.69

Upper H Subscript Upper AHA​:

pnot equals≠0.690.69

B.

Upper H 0H0​:

pequals=0.690.69

Upper H Subscript Upper AHA​:

pless than<0.690.69

C.

Upper H 0H0​:

pequals=0.74400.7440

Upper H Subscript Upper AHA​:

pless than<0.74400.7440

D.

Upper H 0H0​:

pequals=0.690.69

Upper H Subscript Upper AHA​:

pgreater than>0.690.69

E.

Upper H 0H0​:

pequals=0.74400.7440

Upper H Subscript Upper AHA​:

pgreater than>0.74400.7440

F.

Upper H 0H0​:

pequals=0.74400.7440

Upper H Subscript Upper AHA​:

pnot equals≠0.74400.7440

​b) Verify that the conditions are satisfied and find the​ P-value.

Which assumptions and conditions below are​ met? Select all that apply.

A.

The​ success/failure condition is met.

B.

The independence assumption is met.

C.

The​ 10% condition is met.

D.

The randomization condition is met.

What is the​ P-value?

​P-valueequals=nothing

​(Round to four decimal places as​ needed.)

​c) Would you recommend this company based on what you see​ here?

A.The​ P-value provides

insufficient evidenceinsufficient evidence

that the program is successful.

B.The​ P-value provides

some indicationsome indication

that the program is successful.

C.The​ P-value provides

strong evidencestrong evidence

that the program is successful.

D.

Inference is not possible since the assumptions and conditions are not met.

Answer 1

a.)

Correct Option is D

Explanation :

For one sample greater than proportion test the hypothesis to be tested are

H₀ : P = P₀ against H₁ : P > P0

where P₀ is the population proportion .

The hypothesis to be tested here are :

H₀ : P = 0.69

against

H₁ : P > 0.69

As the company wants to test the improvement of the selection process of the company hence the alternative hypothesis will be of greater than type

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b.)

Correct option is  A, B, C and D

P - Value : 0.0324 .

Explanation :

Condition of Independence :

The independence condition our sample size shouldn't exceed 10% of the population size. Since our sample of 250 is less than 10%  of the population, the sample satisfies the independence condition.

Condition of Success / Failure :

In this assumption we expect at least 10 successes and 10 failures in our sample.

Here in our example n = 250 and p = 186/250 = 0.744

There are approximately :

np = 250 * 0.744 = 186 Successes and

n(1-p) = 250*(1-0.744) = 64 failures .

Thus we can conclude that np > 10 and n(1-p) > 10 .

Since both are greater than 10 the success / failure condition is satisfied and verified .

the normal condition is satisfied.

Condition of 10% :

This condition states that the sample sizes should be no more than 10 % of the population:

Here the sample size is 250 of tens of thousands of population .

Hence the 10 % condition is also met

Condition of Randomization :

Since the sample taken is not a biased data mentioned in the example it is a representative of the population , condition of randomization is met .

P - Value :

the z- test statistic here is 1.846 and is computed as follows

The p-value is 0.0324 ..........................................(Calculated from normal probability table Corresponding to 1.846)

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c.)

Correct answer is C.

Explanation :

Decision :

Since it is observed that z = 1.846 zc = 1.64z=1.846>zc​=1.64, it is then concluded that the null hypothesis is rejected.

where zc is the critical region .

Using the P-value approach: The p-value is p =0.0324, and since p=0.0324<0.05, it is concluded that the null hypothesis is rejected and p > 0.69 is accepted .

Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the population proportion  is greater than p0 and we can conclude that the program is successful .

Hence p value provides strong evidence to reject null hypothesis as it is 0.0324 and is less than 0.05.

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Ans :

a.) D

b.) A, B, C, D

c.) C

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