Question

The following data shows the number of hours that 9 hospital patients slept following the administration of a certain anesthetic. 4,8,11,11,5,11,4,8,8 Find a 99% confidence interval for the average hours slept following the administration of the anesthetic for the sampled population. (b) Which of the following statements is true regarding part (a)? contidencer interval Protslerm z(a)= enter your answer in the form 0,6 (rumbers correct to 2 decimals) (A) The population standard deviation σ must be known. (B) The population mean must be inside the confidence interval. (C) The population must be normal. (D) The population does not need to be normal. (E) The population must follow a t-distrabution.

Answer 1

The 99% confidence interval for the average hours slept following the administration of the anesthetic for the sampled population is (5.19, 9.81).

Step-by-step explanation:

To find the 99% confidence interval for the average hours slept, we can use the formula:

`\[ \bar{x} \pm z(a/2) * (s)/(√(n)) \]`

where:

`- \( \bar{x} \)` is the sample mean,

`- \( z(a/2) \)` is the z-score corresponding to the desired confidence level,

`- \( s \)` is the sample standard deviation, and

`- \( n \)` is the sample size.

In this case, the sample mean
`\( \bar{x} \)` is calculated as (4 + 8 + 11 + 11 + 5 + 11 + 4 + 8 + 8) / 9 = 8.

The sample standard deviation
`\( s \)` is approximately 2.36.

The z-score for a 99% confidence interval is 2.92 for a two-tailed test.

Substituting these values into the formula:

`\[ 8 \pm 2.92 * (2.36)/(√(9)) \]`

Calculating this expression results in the 99% confidence interval of (5.19, 9.81).

This means that we are 99% confident that the true average hours slept following the administration of the anesthetic for the sampled population falls within the interval of 5.19 to 9.81 hours. The population does not need to be normal for the confidence interval to be valid, as we are relying on the Central Limit Theorem, which states that the distribution of the sample mean becomes approximately normal as the sample size increases.