the average temp of the lake is 64.2 degrees and the standard deviation is 3.2. Assume the cariable is normally distributed. For a randomly selected day find the probabliity that the temp will be above 62 degrees
Use z-scores:
z = (x - mean)/sd
x = 62
mean = 64.2
sd = 3.2
Once you have the z-score, check a z-table for the probability.
To find the probability that the temperature will be above 62 degrees, we need to use the concept of the standard normal distribution. We can convert the temperature values to z-scores, which measures the number of standard deviations a particular value is from the mean.
First, we need to calculate the z-score for a temperature of 62 degrees. The formula for calculating z-score is:
z = (x - μ) / σ
where z is the z-score, x is the given value, μ is the mean, and σ is the standard deviation.
Using the given values, the calculation becomes:
z = (62 - 64.2) / 3.2
z = -0.688
Next, we need to find the probability of a z-score greater than -0.688. We can refer to a standard normal distribution table or use a calculator with cumulative distribution function (CDF) capabilities.
For this particular question, we want to find the area under the normal curve to the right of the z-score (-0.688). This represents the probability of the temperature being above 62 degrees.
Using a standard normal distribution table or calculator, we find that the probability corresponding to a z-score of -0.688 is approximately 0.752.
Therefore, the probability that the temperature will be above 62 degrees on a randomly selected day is approximately 0.752, or 75.2%.