Questions: Suppose speeds of vehicles traveling on a highway have an unknown distribution with mean 63 and standard deviation 4 miles per hour. A sample of size n=44 is randomly taken from the population and the mean is taken. Using the Central Limit Theorem for Means, what is the standard deviation for the sample mean distribution?

Suppose speeds of vehicles traveling on a highway have an unknown distribution with mean 63 and standard deviation 4 miles per hour. A sample of size n=44 is randomly taken from the population and the mean is taken. Using the Central Limit Theorem for Means, what is the standard deviation for the sample mean distribution?

Transcript text: Suppose speeds of vehicles traveling on a highway have an unknown distribution with mean 63 and standard deviation 4 miles per hour. A sample of size $n=44$ is randomly taken from the population and the mean is taken. Using the Central Limit Theorem for Means, what is the standard deviation for the sample mean distribution?

Solution

Solution Steps

Step 1: Given Information

We are provided with the following parameters regarding the speeds of vehicles traveling on a highway:

Population mean ($ \mu $): 63 mph
Population standard deviation ($ \sigma $): 4 mph
Sample size ($ n $): 44

Step 2: Central Limit Theorem Application

According to the Central Limit Theorem, the distribution of the sample mean will approach a normal distribution as the sample size increases, regardless of the population's distribution. The standard deviation of the sample mean distribution, known as the standard error ($ SE $), can be calculated using the formula:

\[ SE = \frac{\sigma}{\sqrt{n}} \]

Step 3: Calculation of Standard Error

Substituting the known values into the formula:

\[ SE = \frac{4}{\sqrt{44}} \]

Calculating $ \sqrt{44} $:

\[ \sqrt{44} \approx 6.6332 \]

Now, substituting this value back into the equation for $ SE $:

\[ SE \approx \frac{4}{6.6332} \approx 0.6020 \]

Rounding to two decimal places, we find:

\[ SE \approx 0.60 \]