Assume that thermometer readings are normally distributed with a mean of 0 degrees Celsius and a standard deviation of 1 degree Celsius. A thermometer is randomly selected and tested. Find the temperature reading that separates the bottom 40.9% from the top 59.1%. Round your answer to the nearest hundredth.

1 answer

To find the temperature reading that separates the bottom 40.9% from the top 59.1%, we need to determine the z-score corresponding to the cumulative probability of 0.409, since the bottom 40.9% corresponds to a cumulative probability of 0.409 in the standard normal distribution.

We can use a z-table or statistical software to find the z-score.

Using a z-table, we look for the cumulative probability closest to 0.409. Let’s find the z-score:

  • The z-score corresponding to a cumulative probability of 0.409 is approximately -0.23 (since the closest value in the table for 0.409 is 0.4099 which corresponds to -0.23).

Now we use the z-score formula to find the temperature \(X\):

\[ X = \mu + z \cdot \sigma \]

where:

  • \(\mu = 0\) (mean)
  • \(z \approx -0.23\)
  • \(\sigma = 1\) (standard deviation)

Substituting in the values:

\[ X = 0 + (-0.23) \cdot 1 = -0.23 \]

Rounding this to the nearest hundredth, the temperature reading that separates the bottom 40.9% from the top 59.1% is:

\[ \boxed{-0.23} , \text{degrees Celsius} \]