Question

A bottle maker believes that 11% of his bottles are defective. If the bottle maker is right, what is the probability that the proportion of defective bottles in a sample of 529529 bottles would be less than 9%9%

Answer 1

The value is
`P( X < 0.09) = 0.070781`

Explanation:

From the question we are told that

The population proportion is p = 0.11

The sample size is n = 529

Generally given that the sample size is large enough (i.e n > 30), then the mean of this sampling distribution is mathematically represented as

`\mu_(x) = p = 0.11`

Generally the standard deviation of this sampling distribution is

`\sigma = \sqrt{ ( p (1 - p ))/(n) }`

=>
`\sigma = \sqrt{ ( 0.11 (1 - 0.11 ))/(529) }`

=>
`\sigma = 0.01360`

Generally the probability that the proportion of defective bottles in a sample of 529 bottles would be less than 9% (0.09) is mathematically represented as

`P( X < 0.09) = P((X - \mu_(x))/(\sigma) < (0.09 -0.11)/( 0.01360) )`

`(X -\mu)/(\sigma )  =  Z (The  \ standardized \  value\  of  \ X )`

`P( X < 0.09) = P(Z< -1.47 )`

Generally from the z table the area under the normal curve to the left corresponding to -1.47 is

`P(Z< -1.47 ) = 0.070781`

So

`P( X < 0.09) = 0.070781`