Suppose you think that having social media followers is simply a popularity cont

Question

Suppose you think that having social media followers is simply a popularity contest. You think that the prettier a person is, the more followers they will have. You decide to test this by interviewing, 500 people. You ask them how many followers they have. Then you rate their physical appearance on a scale between 1 to 100, where 1 represents an extremely ugly person. You next run a regression on the following model: Followers_i = beta_1 + beta_2 Beauty_i +u_i. You find that b_1 = 30, s.c.(b_1) = 11, b_2 = 7.28, and s.c. (b_2) = 2.54, Test whether Beta_2 = 10 at the 3% significance level, Your answer should include your test statistic, the critical value, and the result of the test. Can we say that Beta= 10? Test the null hypothesis that the effect of beauty on the number of followers is non-positive at the 1% level. Your answer should include your test statistic, the critical value, and the result of the test. Can we say that this effect is non-positive? Create a 99% (two-tailed) confidence interval for Beta_2.

Explanation / Answer

Solution

Part (a)

Hypotheses:

Null H0: 2 = 20 = 10   Vs

Alternative HA: 2 10

Test statistic:

t = (b2 - 20)/SE(b2) = (7.28 – 10)/2.54 = - 2.72/2.54 = - 1.0709

Distribution, Critical Value

Under H0, t ~ tn – 2, where n = sample size = 580

Critical value = upper (/2)% point of t578.

Given = 0.05, tcrit = 1.96 [using Excel Function]

Decision Criterion (Rejection Region)

Reject H0, if | tcal | > tcrit

Decision:

Since | tcal | < tcrit,

H0 is accepted.

Conclusion:

There is sufficient evidence to support the claim that 2 = 10. ANSWER

Part (b)

Claim: Effect of beauty on the number of followers is non-positive

Hypotheses:

Null H0: 2 = 20 = 0   Vs

Alternative HA: 2 > 0

Test statistic:

t = (b2 - 20)/SE(b2) = (7.28 – 0)/2.54 = 7.28/2.54 = 2.8661

Distribution, Critical Value

Under H0, t ~ tn – 2, where n = sample size = 580

Critical value = upper % point of t578.

Given = 0.01, tcrit = 2.576 [using Excel Function]

Decision Criterion (Rejection Region)

Reject H0, if tcal > tcrit

Decision:

Since tcal > tcrit,

H0 is rejected.

Conclusion:

H0 is rejected => HA is accepted => 2 > 0 =>There is sufficient evidence to support the claim that 2 is NOT non-positive. ANSWER

Part (c)

99% confidence interval for 2 is given by b2 ± SE(b2) x tn – 2,/2, where = 0.01.

= 7.28 ± (2.54 x 2.326) = 7.28 ± 5.91

Lower Bound: 1.37 and Upper Bound: 13.19 ANSWER

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