Final answer:
To answer this question, we use the concept of z-scores to determine the number of springs that can withstand specific forces. While statements a and b are true, statement c is not, as it overestimates the number of springs that can withstand more than 206 lbs. Statement d is also false, as it underestimates the number of springs that can withstand less than 194 lbs.
Step-by-step explanation:
To answer this question, we can use the concept of z-scores. The z-score measures how many standard deviations an observation is from the mean. In this case, the mean force that the springs can withstand is 200 lbs, with a standard deviation of 6 lbs. Let's look at each option:
- a. About 42 springs can withstand 218 lbs of force or less.
This is true because we can calculate the z-score for 218 lbs using the formula: z = (x - mean) / standard deviation = (218 - 200) / 6 = 3.
According to the standard normal distribution table, the area to the left of a z-score of 3 is approximately 0.9987. So, about 0.9987 or 99.87% of springs can withstand 218 lbs of force or less. - b. About 650 springs could withstand between 194 and 206 lbs of force.
This is also true because we can calculate the z-scores for 194 lbs and 206 lbs and find the corresponding areas under the normal curve. The z-score for 194 lbs is (194 - 200)/6 = -1, and the z-score for 206 lbs is (206 - 200)/6 = 1.
Using the standard normal distribution table, we can find that the area to the left of a z-score of -1 is approximately 0.1587, and the area to the left of a z-score of 1 is approximately 0.8413.
Therefore, the area between a z-score of -1 and 1 is approximately 0.8413 - 0.1587 = 0.6826.
So, about 0.6826 or 68.26% of springs can withstand between 194 and 206 lbs of force. - c. About 130 springs could withstand more than 206 lbs of force.
This statement is not true because we can calculate the z-score for 206 lbs (as shown above) and find the corresponding area to the left of the z-score, which is 0.8413.
This means that about 84.13% of springs can withstand 206 lbs of force or less.
Therefore, 100% - 84.13% = 15.87% of springs can withstand more than 206 lbs of force, not 130 springs. - d. About 25 springs could withstand less than 194 lbs of force.
This statement is also not true because we can calculate the z-score for 194 lbs (as shown above) and find the corresponding area to the left of the z-score, which is 0.1587.
This means that about 15.87% of springs can withstand 194 lbs of force or less, not 25 springs.
In conclusion, the statement that is not true is c. About 130 springs could withstand more than 206 lbs of force.