Please could you explain your answers (for the benefit of me) and your formulas.

Question

Please could you explain your answers (for the benefit of me) and your formulas. Trying to really understand this topic.

Single proportions:

1. You have run a preference test where 70% of the participants preferred product “A” over product “B”. If the number of participants was 15, is there a statistical difference? (i.e., was there in fact a preference for product “A”?).

2. For the 70% preference (for product “A”), how many participants did you need in your study to achieve 0.05 significance (95% confidence)?

3. For your original group of participants (n = 15), what is the % preference that you would need to observe to achieve statistical significance (using 0.05 significance)?

Two proportions:

4. You have an “older subject” group and a “younger subject” group. For the older group, 7 out of 8 preferred product “A”. For the younger group 8 out of 12 preferred product “A”. Is there a statistical difference between the two groups? (do “older” subjects prefer product “A” more than the “younger” subject group?)

5. You have an “older subject” group and a “younger subject” group. For the older group, 15 out of 20 preferred product “A”. For the younger group 4 out of 13 preferred product “A”. Is there a statistical difference between the two groups? (do “older” subjects prefer product “A” more than the “younger” subject group?)

6. You have an “older subject” group and a “younger subject” group. For the older group, 75 out of 100 preferred product “A”. For the younger group 45 out of 76 preferred product “A”. Is there a statistical difference between the two groups? (do “older” subjects prefer product “A” more than the “younger” subject group?)

Explanation / Answer

Question 1

Yes, there is a statistical difference. There is a more preference for product A. Preference for product A is 70% and preference for product B would be remaining 30%.

Question 2

We are given,

Proportion = p = 0.70,

Sampling error or margin of error = E = 0.05,

Confidence level = c = 0.95 or 95%

Sample size = n = p*q*(Z/E)^2

Where, q = 1 – p = 1 – 0.70 = 0.30

Z = 1.96 (by using z-table)

n = p*q*(Z/E)^2

n = 0.70*0.30*(1.96/0.05)^2

n = 322.6944

Required participants = 323

Question 3

n = p*q*(Z/E)^2

We are given n = 15, E = 0.05, Z = 1.96

15 = p*(1 – p)*(1.96/0.05)^2

p*(1 – p) = 15/(1.96/0.05)^2 = 0.009762

p – p^2 - 0.009762 = 0

p^2 – p + 0.009762 = 0

(p – 0.990141)(p – 0.0098592) = 0

p = 0.99 approximately

% of preference = 99%

Question 4

We are given

For older group, X=7, n = 8, p = 7/8 = 0.875

For younger group, X = 8, n = 12, p = 8/12 = 0.667

Here, we have to check whether there is any statistically significant difference in the proportions for older group and younger group.

For checking this claim or hypothesis we have to use z test for two population proportions. The null and alternative hypothesis for this test is given as below:

H0: polder = pyounger

Ha: polder pyounger

We assume 5% level of significance for this test.

Test statistic formula for this test is given as below:

Z = (polder – pyounger) / sqrt[(polder(1 – polder)/nolder)+( pyounger(1 – pyounger)/nyounger)]

Z = (0.875 - 0.667)/sqrt(0.875*(1 - 0.875)/8)+(0.667*(1 - 0.667)/12))

Z = 1.7974

p-value = 0.0723

P-value > alpha = 0.05

So, we do not reject the null hypothesis

There is insufficient evidence to conclude that there is a statistical difference between the two groups.

Related Questions

Navigate

• Browse (All)
• Browse P
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.