QuestionThe mean age of all 627 used cars for sale in a newspyor one Saturday last month was 7.8 years, with a stardard deviation of 7.6 years. The distribution of agns is right-skened age of the 40 cars he samples is 8.4 years and the standard deviation of those 40 cars is 5.8 years. Complete parts a through c .
(type integers or occimals.)
c. Are the conditions for using the CLT (Central Limit Theorem) fulfilled?
A. No, because the Normal condition is not fulfilled.
B. No, because the random sample/independence and Normal conditions are not fulfilled.
C. No, because the random samplelindependence condition is not fulfilled.
D. Yes, all the conditions for using the CLT are fulfilled.
What would be the shape of the approximate sampling distribution of a large number of means, each from a sample of 40 cars?
Normal
Rinht-clement
Studdy Solution
STEP 1
What is this asking? Can we use the Central Limit Theorem (CLT) with this car age data, and if so, what would the distribution of sample means look like? Watch out! Don't confuse the distribution of the *car ages* with the distribution of the *sample means* of car ages!
STEP 2
1. Check the Random Sample/Independence condition.
2. Check the Normal condition.
3. Determine the shape of the sampling distribution.
STEP 3
We're told the 40 cars are a sample.
We need to consider if this sample is random and independent.
A truly random sample would mean each car has an equal chance of being selected, and the selection of one car doesn't influence the selection of another.
We don't have enough information to confirm true randomness and independence, so we'll *assume* it's reasonably random and proceed with caution!
STEP 4
The CLT usually needs either the original population to be normally distributed *or* a large enough sample size.
Our population (all 627 cars) is right-skewed, *not* normal.
STEP 5
So, we look to the sample size.
A sample size of 40 is generally considered large enough for the CLT to apply, even if the original population isn't normal.
This is because the CLT states that as the sample size increases, the distribution of the sample means approaches a normal distribution regardless of the shape of the original population.
STEP 6
Since our sample size is large enough (), the CLT kicks in!
This tells us that the sampling distribution of the means of many samples of 40 cars will be approximately normal, *even though* the original car ages are right-skewed.
STEP 7
c. The answer is **D**.
Yes, all the conditions for using the CLT are fulfilled (assuming a reasonably random sample).
The sample size is large enough to overcome the right-skewed population distribution.
The shape of the approximate sampling distribution of a large number of means, each from a sample of 40 cars, would be **Normal**.
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