Question

The standard deviation of a sample of the costs of 15 automobile repairs at a local garage was $ 120.82 . Assuming that the variable is normally distributed, find the critical values,

Answer 1

To find the critical values for a 95% confidence interval of the sample mean cost of automobile repairs, use the Z-table or software with a Z-score of approximately ±1.96, multiplying by the standard error of the mean (SEM).

Step-by-step explanation:

To find the critical values for a normally distributed variable, you typically refer to the Z-table or use statistical software. The critical values correspond to the Z-scores that define the tails of the distribution, indicating the range within which a certain percentage of the data falls.

Since the standard deviation of the sample costs is given as $120.82, you'll need to use this information to calculate the standard error of the mean (SEM), which is given by the formula:

`\[SEM = (s)/(√(n))\]`

where
`\(s\)` is the sample standard deviation and
`\(n\)` is the sample size. In this case,
`\(s = 120.82\)` and
`\(n = 15\)`.

`\[SEM = (120.82)/(√(15))\]`

After calculating the SEM, you can use it to find the critical values based on the desired level of confidence (usually 90%, 95%, or 99%). The critical values correspond to the Z-scores that cut off the tails of the normal distribution.

For example, if you want to find the critical values for a 95% confidence interval, you'll look for the Z-scores that leave 2.5% in each tail. You can then use these Z-scores to find the corresponding raw values by multiplying them by the SEM and adding/subtracting from the sample mean.