The amount of time full-time employed Americans spend daily on working and work related activities is normally distributed with a mean of 6.27 hours and a standard deviation of 0.52 hours. What proportion of full-time employed Americans spend less than 7.5 hours daily on working and work related activities? The z-score is and the proportion is Type your answer here.
The correct answer and explanation is :
To determine the proportion of full-time employed Americans who spend less than 7.5 hours daily on work-related activities, we can utilize the properties of the normal distribution. Here’s a step-by-step explanation:
1. Understanding the Normal Distribution Parameters:
The daily work hours are normally distributed with:
- Mean (μ): 6.27 hours
- Standard Deviation (σ): 0.52 hours
2. Calculating the Z-Score:
The Z-score represents how many standard deviations a specific value (X) is from the mean. It’s calculated using the formula:
[ Z = \frac{X – \mu}{\sigma} ]
Substituting the given values:
[ Z = \frac{7.5 – 6.27}{0.52} \approx 2.37 ]
3. Determining the Proportion Using the Z-Score:
A Z-score of 2.37 corresponds to the area under the standard normal curve to the left of this value. Using standard normal distribution tables or computational tools, we find that the cumulative probability for a Z-score of 2.37 is approximately 0.9910.
4. Interpreting the Result:
This means that approximately 99.10% of full-time employed Americans spend less than 7.5 hours daily on work-related activities.
Explanation:
- The peak of the curve represents the mean (6.27 hours).
- The area under the curve to the left of the Z-score (2.37) encompasses approximately 99.10% of the data.
- The tail to the right represents the remaining 0.90% of individuals who spend more than 7.5 hours daily on work-related activities.
Conclusion:
Utilizing the properties of the normal distribution, we determined that a vast majority (approximately 99.10%) of full-time employed Americans spend less than 7.5 hours daily on work-related activities.