Question

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 +1σ point, which is 19.83. What is the correct conclusion? (1 point) Responses 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 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 +1σ point, and the model predicts 68%. The normal model is a good fit because 85% of the data are less than the value at the plus 1 sigma point, and the model predicts 68%. 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 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 +1σ point, and the model predicts 84%. The normal model is a good fit because 85% of the data are less than the value at the plus 1 sigma point, and the model predicts 84%. Skip to navigation

Answer 1

To evaluate Lydia's application of the Empirical Rule to her collected data, let's first summarize the important points:

1. Lydia's data consists of the number of cars passing by over a 5-minute period.
2. She determined that the normal model, which approximates this data, is \( N(16.05, 3.78) \).
3. The mean of this dataset is approximately 16.05, and the standard deviation is roughly 3.78.
4. The one standard deviation above the mean (\(+1σ\)) can be calculated as: \[ 16.05 + 3.78 \approx 19.83 \]

Under the Empirical Rule (68-95-99.7 Rule), we can predict that approximately 68% of the data should fall within one standard deviation (from \( \mu - \sigma \) to \( \mu + \sigma \)).

Now, according to Lydia, she found that:

• 85% of the data is less than the value at the +1σ point (19.83).
• Under the normal model, we would expect approximately 68% of the data to fall within one standard deviation from the mean.

Now we evaluate the statements:

1. The normal model is not a good fit because 45% of the data are less than the mean, and the model predicts 50%.

   • (This focuses on the left side of the distribution, but the main concern here is the proportion of data to the right of the +1σ point.)

2. 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 68%.

   • (This statement presents a reasonable conclusion based on the observation that significantly more data is below the +1σ value than expected, indicating potential skewness or non-normality.)

3. 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%.

   • (Similar reasoning applies here, though the prediction of 84% differs from the standard expectation of 68%. Still, it acknowledges a good fit as it implies a significant portion is below that threshold.)

Based on the 85% of data being below +1σ and the Empirical Rule predicting 68%, the conclusion should be that the model is not a perfect fit. The significant difference indicates that the actual data diverges from the normal distribution prediction for that \( +1σ \) point.

Considering the statements provided, it seems that:

• The most suitable conclusion is that the normal model is not a good fit since it translates directly to the data's distribution characteristics not aligning with what the normal model predicts, particularly regarding the number against the Empirical Rule specifics at the +1σ point.

The correct response should be: The normal model is not a good fit because 85% of the data are less than the value at the +1σ point, while the model predicts 68%. This indicates that the actual distribution does not adhere to the typical characteristics of a normal distribution.