Yes, because the upper limit of the 68.26% interval is less than 8 minutes.
(d) How do the percentages of the 100 waiting times in Table 1.9 that actually fall into intervals [$\bar{x} \pm s$], [$\bar{x} \pm 2 s$], and [$\bar{x} \pm 3 s$] compare to those given by the Empirical Rule? Do these comparisons indicate that the statistical inferences you made in parts b and c are reasonably valid? (Round your answers to the nearest whole number.)
Answer (2)
The Empirical Rule states that approximately 68% of data falls within 1 standard deviation, 95% within 2, and 99.7% within 3.
The sample data shows 67% within 1 standard deviation, 97% within 2, and 100% within 3.
These percentages are close to the Empirical Rule, with differences of 1%, 2%, and 0.3% respectively.
Therefore, the statistical inferences are reasonably valid. Y es
Explanation
Understand the problem We need to compare the given percentages with the Empirical Rule and determine if the statistical inferences made in parts b and c are reasonably valid.
State the Empirical Rule The Empirical Rule states that approximately 68% of the data falls within one standard deviation of the mean ( [ x ˉ \tpm s ] ), 95% falls within two standard deviations ( [ x ˉ \tpm 2 s ] ), and 99.7% falls within three standard deviations ( [ x ˉ \tpm 3 s ] ).
List given percentages The given percentages from the sample data are: 67% within [ x ˉ \tpm s ] , 97% within [ x ˉ \tpm 2 s ] , and 100% within [ x ˉ \tpm 3 s ] .
Compare the percentages Comparing the given percentages with the Empirical Rule:
For [ x ˉ \tpm s ] : 67% (sample) vs. 68% (Empirical Rule). The difference is 1%.
For [ x ˉ \tpm 2 s ] : 97% (sample) vs. 95% (Empirical Rule). The difference is 2%.
For [ x ˉ \tpm 3 s ] : 100% (sample) vs. 99.7% (Empirical Rule). The difference is 0.3%.
Assess validity of inferences The sample percentages are very close to the Empirical Rule percentages. The small differences suggest that the statistical inferences made in parts b and c are reasonably valid.
Final Answer Yes, the statistical inferences made in parts b and c are reasonably valid.
Examples
In quality control, the Empirical Rule helps determine if a manufacturing process is stable. If the percentage of products falling within certain standard deviations of the mean deviates significantly from the Empirical Rule, it indicates a problem in the process that needs investigation. For example, if a machine is supposed to produce bolts with a mean length of 5 cm and a standard deviation of 0.1 cm, the Empirical Rule suggests that about 95% of the bolts should have lengths between 4.8 cm and 5.2 cm. If a sample shows a significantly different percentage, the machine may need recalibration.
The comparisons reveal that the sample percentages (67% within 1 standard deviation, 97% within 2, and 100% within 3) are very close to those predicted by the Empirical Rule (68%, 95%, and 99.7% respectively). The small differences suggest that the statistical inferences from parts b and c are reasonably valid. Therefore, the answer is yes.
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