Question

In a recent year, 6% of cars sold had a manual transmission. A random sample of college students who owned cars revealed the following: out of 124 cars, 29 had manual transmissions. Estimate the proportion of college students who drive cars with manual transmissions with 90% confidence.

Answer 1

To estimate the proportion of college students who drive cars with manual transmissions with 90% confidence, a confidence interval can be used. The proportion falls between 0.1794 and 0.2896.

Step-by-step explanation:

To estimate the proportion of college students who drive cars with manual transmissions with 90% confidence, we can use a confidence interval.

Given that 29 out of 124 cars in the sample had manual transmissions, the sample proportion is 29/124 ≈ 0.2339.

Using the formula for the confidence interval for a proportion, the lower bound is approximately 0.1794 and the upper bound is approximately 0.2896. Therefore, we can say with 90% confidence that the proportion of college students who drive cars with manual transmissions is between 0.1794 and 0.2896.

Answer 2

The estimate the proportion of college students who drive cars with manual transmissions with 90% confidence is

`0.171 <  p <  0.297`

Step-by-step explanation:

From the question we are told that

The sample size is n = 124

The population proportion p = 0.06

The number of cars with manual transmissions is k = 29

From the question we are told the confidence level is 90% , hence the level of significance is

`\alpha = (100 - 90 ) \%`

=>
`\alpha = 0.10`

Generally the sample proportion is mathematically represented as

`\^ p = ( 29 )/(124 )`

=>
`\^ p = 0.234`

Generally from the normal distribution table the critical value of
`(\alpha )/(2)` is

`Z_{(\alpha )/(2) } =  1.645`

Generally the margin of error is mathematically represented as

`E =  Z_{(\alpha )/(2) } * \sqrt{(\^ p (1- \^ p))/(n) }`

=>
`E =   1.645  * \sqrt{(0.234 (1- 0.234 ))/(124) }`

=>
`E =  0.063`

Generally 95% confidence interval is mathematically represented as

`\^ p -E <  p <  \^ p +E`

=>
`0.234  - 0.063  <  p <  0.234  + 0.063`

=>
`0.171 <  p <  0.297`