Question

An institute reported that 60​% 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.

b. The probability is 90​% that the sample percentage will be contained within what symmetrical limits of the population​ percentage? The probability is 90​% that the sample percentage will be contained above nothing​% and below nothing​%.

Answer 1

Given that,

p = 60% = 0.60

1 - p = 1 - 0.60 = 0.40

n = 100

= p = 0.60

=  [p( 1 - p ) / n] = [(0.60 * 0.40) / 100] = 0.0490

Using standard normal table,

P( -z < Z < z) = 90%

= P(Z < z) - P(Z <-z ) = 0.90

= 2P(Z < z) - 1 = 0.90

= 2P(Z < z) = 1 + 0.90

= P(Z < z) = 1.90 / 2

= P(Z < z) = 0.95

= P(Z < 1.645) = 0.95

= z  ± 1.645

Using z-score formula,

  = z *   +  

  = -1.645 * 0.0490 + 0.60

  = 0.5194

  = 51.94%

Using z-score formula,

  = z *   +  

  = 1.645 * 0.0490 + 0.60

  = 0.681

  = 68.1%

The probability is 90% that the sample percentage will be contained above 51.94 % and below 68.1 %