Question

Tesla wanted to determine the average miles per kWh that their vehicles get across all models and variations. They took a sample of 100 of their vehicles and recorded the resulting data. Assume that the true population mean is 30 kWh and that the population standard deviation is 3. Determine the correct mean and standard deviation for the sampling distribution of the sample mean__________ The mean is and the standard deviation is __________

Answer 1

`\mu_(\bar x) = \mu = E(X) =30KWh`

`\sigma_(\bar X)= (\sigma)/(√(n))=(3)/(√(100))=0.3`

Explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".

The central limit theorem states that "if we have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed. This will hold true regardless of whether the source population is normal or skewed, provided the sample size is sufficiently large".

Solution to the problem

For this case we select a sample of n =100

From the central limit theorem we know that the distribution for the sample mean
`\bar X` is given by:

`\bar X \sim N(\mu, (\sigma)/(√(n)))`

So then the sample mean would be:

`\mu_(\bar x) = \mu = E(X) =30KWh`

And the standard deviation would be:

`\sigma_(\bar X)= (\sigma)/(√(n))=(3)/(√(100))=0.3`