How do I solve the question below?

Question

Assume that the readings at freezing on a batch of thermometers are normally distributed with a mean of 0°C and a standard deviation of 1.00°C. A single thermometer is randomly selected and tested. Find P₅₈, the 58-percentile. This is the temperature reading separating the bottom 58% from the top 42%.

P₅₈ = °C

(Round answer to three decimal places)

Answer 1

To find P₅₈, we need to determine the temperature reading that separates the bottom 58% from the top 42% of thermometer readings.

Using the standard normal distribution table or a calculator with a normal distribution function, we can find the z-score corresponding to the 58th percentile:

z = InvNorm(0.58) ≈ 0.21

where InvNorm is the inverse normal cumulative distribution function.

The z-score represents the number of standard deviations away from the mean for a given percentile. To find the corresponding temperature reading, we use the formula:

z = (x - μ) / σ

where x is the temperature reading we want to find, μ is the mean temperature (0°C), and σ is the standard deviation (1.00°C). Solving for x:

x = z * σ + μ x = 0.21 * 1.00 + 0 x = 0.21

Therefore, the temperature reading at P₅₈ is 0.21°C.