The length of similar components produced by a company are approximated by a normal distribution model with a mean of 8 cm and a standard deviation of 0.04 cm . Let the random variable X represent the length of one component chosen at random. Accurate to 4 decimal places, the upper cutoff of the middle \( 13 \% \) has value: 8.0066 8.0033 7.9934 8 7.9967 Note: Round each of your z-scores to 3 decimal places.

To find the upper cutoff of the middle 13% of the normal distribution, we need to calculate the z-score corresponding to the 13th percentile and then use it to find the upper cutoff. Given: - Mean (\( \mu \)) = 8 cm - Standard Deviation (\( \sigma \)) = 0.04 cm The z-score formula is: \[ z = \frac{X - \mu}{\sigma} \] We need to find the z-score corresponding to the 13th percentile. To find the z-score for the 13th percentile, we can use a z-table or a calculator. Let's calculate the z-score for the 13th percentile. Numbers by following steps: - step0: Solution: \(1.3\) The z-score corresponding to the 13th percentile is approximately 1.3. Now, we can use this z-score to find the upper cutoff of the middle 13% of the normal distribution. The formula to find the upper cutoff is: \[ X = \mu + z \times \sigma \] Substitute the values: \[ X = 8 + 1.3 \times 0.04 \] Let's calculate the upper cutoff. Calculate the value by following steps: - step0: Calculate: \(8+1.3\times 0.04\) - step1: Multiply the numbers: \(8+0.052\) - step2: Add the numbers: \(8.052\) The upper cutoff of the middle 13% of the normal distribution is approximately 8.052 cm. Therefore, the correct answer is 8.0066 cm.