Question

Please give me detailed formula. I keep calculating the wrong way. An institute reported that 65​% of its members indicate that lack of ethical culture within financial firms has contributed most to the lack of trust in the financial industry. Suppose that you select a sample of 100 institute members.

Complete parts ​(a) through ​(d) below.

a. What is the probability that the sample percentage indicating that lack of ethical culture within financial firms has contributed the most to the lack of trust in the financial industry will be between 58​% and 73​%? ​____ and ______(Type an integer or decimal rounded to four decimal places as​ needed.)

b. The probability is 60​% that the sample percentage will be contained within what symmetrical limits of the population​ percentage? The probability is 60​% that the sample percentage will be contained above ______​% and below _______%. ​(Type integers or decimals rounded to one decimal place as​ needed.)

Answer 1

Solution

Given that,

p = 0.65

1 - p = 1 - 0.65 = 0.35

n = 100

= p = 0.65

=  [p ( 1 - p ) / n] = [(0.65 * 0.35) / 100 ] = 0.0477

a) P( 0.58 < < 0.73 )

= P[(0.58 - 0.65) / 0.0477 < ( - ) / < (0.73 - 0.65) / 0.0477]

= P(-1.47 < z < 1.68)

= P(z < 1.68) - P(z < -1.47)

Using z table,   

= 0.9535 - 0.0708

=0.8827

b) Using standard normal table,

P( -z < Z < z) = 60%

= P(Z < z) - P(Z <-z ) = 0.60

= 2P(Z < z) - 1 = 0.60

= 2P(Z < z) = 1 + 0.60

= P(Z < z) = 1.60 / 2

= P(Z < z) = 0.80

= P(Z < 0.84) = 0.80

= z  ± 0.84

Using z-score formula,

  = z *   +  

  = -0.84 * 0.0477 + 0.65

  = 0.610

  = 61.0%

Using z-score formula,

  = z *   +  

  = 0.84 * 0.0477 + 0.65

  = 0.690

  = 69.0%

The probability is 60% that the sample percentage will be contained above 61.0% and below 69.0%