Question

Life in hours of a 75-watt light bulb is known to be normally distributed with o = 25 hours. A random

sample of 20 bulbs hms a mean life of x 1014 hours.
(a) Construct a 95% two-sided confidence interval on the mean life. Use a = 0.05.
(b) Construct a bound on the mean life. Compare the lower bound of this
confidence interval with the one in part (a).
(c) Suppose that you wanted the total width of the two-sided confidence interval on mean life to be six
hours at 95% confidence. What sample size should be used?

Answer 1

To construct a 95% two-sided confidence interval on the mean life of the 75-watt light bulb, calculate the standard error, margin of error, and lower and upper bounds of the confidence interval. The lower bound of the confidence interval is a bound on the mean life. To determine the sample size needed for a specific total width of the confidence interval, use the formula involving the critical value, margin of error, and desired width.

Step-by-step explanation:

To construct a 95% two-sided confidence interval on the mean life of the 75-watt light bulb, we can use the formula:

1. 
   
3. Calculate the standard error using the formula:
4. 
   

• 
  
• Standard Error = Standard Deviation / sqrt(sample size)
• 
  

Calculate the margin of error using the formula:

• 
  
• Margin of Error = Critical Value * Standard Error
• 
  

Calculate the lower and upper bounds of the confidence interval using the formula:

• 
  
• Lower Bound = Sample Mean - Margin of Error
• 
  
• Upper Bound = Sample Mean + Margin of Error
• 
  

In this case, the critical value for a 95% confidence level is approximately 1.96. Given the sample mean of 1014 hours, standard deviation of 25 hours, and sample size of 20, we can calculate:

1. 
   
3. Standard Error = 25 / sqrt(20) = 5.59
4. 
   
6. Margin of Error = 1.96 * 5.59 = 10.95
7. 
   
9. Lower Bound = 1014 - 10.95 = 1003.05
10. 
    
12. Upper Bound = 1014 + 10.95 = 1024.95

So, the 95% confidence interval on the mean life of the 75-watt light bulb is approximately 1003.05 hours to 1024.95 hours.

To construct a bound on the mean life, we can use the lower bound of the confidence interval. In this case, the lower bound is 1003.05 hours, which is less than the lower bound of the confidence interval in part (a).

To determine the sample size needed for a total width of six hours at a 95% confidence level, we can use the formula:

• 
  
• Sample Size = (Z / Margin of Error) ^ 2

Where Z is the critical value for the desired confidence level. Rearranging the formula, we have:

• 
  
• Margin of Error = Z / sqrt(Sample Size)
• 
  
• 6 = Z / sqrt(Sample Size)
• 
  
• Solving for Sample Size,
• 
  
• Sample Size = (Z / 6) ^ 2

Since we want a total width of six hours at a 95% confidence level, we can use the critical value of 1.96. Plugging in these values, we have:

• 
  
• Sample Size = (1.96 / 6) ^ 2 = 0.653

Therefore, a sample size of approximately 0.653 is required to achieve a total width of six hours at a 95% confidence level.