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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 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+10 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 +10 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%.

1 answer

To evaluate the fit of the normal model \( N(16.05, 3.78) \) to Lydia's data using the Empirical Rule, we first need to analyze the data and see how well it aligns with the predictions made by the normal model.

  1. Empirical Rule Overview: According to the Empirical Rule:

    • About 68% of the data falls within one standard deviation of the mean (i.e., between \( \mu - \sigma \) and \( \mu + \sigma \)).
    • About 95% of the data falls within two standard deviations.
    • About 99.7% falls within three standard deviations.

  2. Data Analysis:

    • Lydia's sample mean \( \mu \) is approximately 16.05.
    • The standard deviation \( \sigma \) is approximately 3.78.

    Calculating the 1-sigma interval:

    • Lower bound: \( 16.05 - 3.78 \approx 12.27 \)
    • Upper bound: \( 16.05 + 3.78 \approx 19.83 \)

  3. Data Checking:

    • We need to calculate how many of the data points fall below \( 19.83 \).
    • The data points are: 17, 8, 17, 21, 16, 15, 13, 10, 18, 17, 23, 17, 18, 10, 18, 13, 16, 21, 17, 16.

    The count of data points less than \( 19.83 \):

    • 17, 8, 17, 16, 15, 13, 10, 18, 17, 18, 10, 18, 13, 16, 17, 16. (15 counts)
    • The data points which are 21 and 23 are above 19.83.

    Therefore, there are 15 data points that are less than 19.83 out of a total of 20.

  4. Percentage Calculation:

    • \( \frac{15}{20} = 0.75 \) or 75% of data points are below \( 19.83 \).

  5. Conclusion Evaluation:

    • The normal model predicts that approximately 68% of the observations should fall within one standard deviation from the mean, but Lydia found that 75% are below \( 19.83 \).
    • This discrepancy means the model slightly underestimates the proportion of data below this point.

  6. Evaluating the Statements:

    • The first and fourth statements suggest that the normal model isn't a good fit because the mean's percentage doesn't align.
    • The second statement claims 85%, which is incorrect based on the analysis since it is around 75%.
    • The third statement also suggests 85% but claims the model predicts 84%, which again is incorrect based on the normal model's expectations.

Given this analysis, the best fit for the conclusion is that the normal model is not a good fit because 45% of the data are less than the mean, and the model predicts 50%.