Question

5. Myers-Briggs: Actors Isabel Briggs Myers was a pioneer in the study of personality types. The following information is taken from MBTI Manual: A Guide to the Development and Use of the Myers-Briggs Type Indicator by Myers and McCaulley (Consulting Psychologists Press). In a random sample of 62 professional actors, it was found that 39 were extroverts.

1. Let p represent the proportion of all actors who are extroverts. Find a point estimate for p.
2. Find a 95% confidence interval for p. Give a brief interpretation of the meaning of the confidence interval you have found.
3. Check Requirements Do you think the conditions np > 5 and nq > 5 are satisfied in this problem? Explain why this would be an important consideration.

Answer 1

Answer:

a)

To give the point estimate

Given,

n = 62

x = 39

p^ = x/n

= 39/62

p^ = 0.63

b)

To give 95% confidence interval

Here for the 95% confidence interval , z value is 1.96

Now consider,

Interval = p^ +/- z*sqrt(p^(1-p^)/n)

substitute values

= 0.63 +/- 1.96*sqrt(0.63(1-0.63)/62)

= 0.63 +/- 0.12

= (0.63 - 0.12 , 0.63 + 0.12)

= (0.51 , 0.75)

Interval = (0.51 , 0.75)

Here we are 95%confident that the interval is one of the interval that contains the unknown value of the p.

c)

Let us consider,

n = 62

p = 0.63

q = 1 - 0.63

q = 0.37

Now,

np = 63*0.63

= 39.69 > 5

nq = 63*0.37

= 23.31 > 5

So the conditions are satisfied

Both np & nq are greater than 5  

It is an important consideration due to that if it is not satisfied then we are not supposed to use the normal approximation method. And the normal curve wouldn't gives the good approximation to binomial.