Question

The lengths of manufactured nails are distributed normally, with a mean length of 6cm, which has a standard deviation of 2mm. what is the length for which 98% of the nails will be longer?

Answer 1

The length for which 98% of the nails will be longer is approximately 6.466 cm.

Explanation:

First, we need to convert the units of the standard deviation to centimeters, since the mean is also given in centimeters. 2 mm is equal to 0.2 cm.

Next, we need to find the z-score that corresponds to the 98th percentile. We can use a standard normal distribution table or calculator to find this value. The z-score corresponding to the 98th percentile is approximately 2.33.

Finally, we can use the formula for a z-score to find the length of nail corresponding to this z-score:

• z = (x - μ) / σ

where:

• z = 2.33
• μ = 6 cm
• σ = 0.2 cm

Solving for x, we get:

• 2.33 = (x - 6) / 0.2
• x - 6 = 0.2 * 2.33
• x - 6 = 0.466
• x = 6.466

Therefore, the length for which 98% of the nails will be longer is approximately 6.466 cm.