Question

modeling with normal distributions quick check
lydia collected samples to see how many cars passed by her house in a period of 5 minutes. this is the data that she collected:
17, 8, 17, 21, 16, 15, 13, 10, 18, 17, 23, 17, 18, 10, 18, 13, 16, 21, 17, 16.
the normal model that lydia calculated for this model is ( n(16.05, 3.78) )
then, lydia used the empirical rule to check whether this data fits the empirical rule. she checked the number of data that were to the left of the ( +1sigma ) point, which is 19.83. what is the correct conclusion?
the empirical rule

(1 point)
the normal model is not a good fit because 45% of the data are less than the mean, and the model predicts 50%.
the normal model is a good fit because 65% of the data are less than the value at the ( +1sigma ) point, and the model predicts 84%.
the normal model is not a good fit because 45% of the data are less than the mean, and the model predicts 50%.
the normal model is a good fit because 85% of the data are less than the value at the ( +1sigma ) point, and the model predicts 68%.

Explanation:

Step1: Count the total number of data points

The data set is \(17,8,17,21,18,15,13,10,18,17,23,17,18,10,18,13,16,21,17,16\). Counting them, we have \(n = 20\) data points.

Step2: Count the number of data points less than the \(+1\sigma\) point (\(19.83\))

Sort the data: \(8,10,10,13,13,15,16,16,17,17,17,17,17,18,18,18,18,21,21,23\).
The number of data points less than \(19.83\) is \(17\).
The proportion is \(\frac{17}{20}=0.85 = 85\%\)
According to the empirical rule, for a normal distribution \(N(\mu,\sigma)\), the proportion of data to the left of \(\mu+\sigma\) is \(50\%+34\%=84\%\approx85\%\)

Answer:

The normal model is a good fit because 85% of the data are less than the value at the + 1σ point, and the model predicts 84%.