At a ski area in Vermont, the daytime high temperature is normally distributed during
January, with a mean of 22°F and a standard deviation of 10°F. You are planning to
snowboard there this January. What is the probability that you will encounter daytime
highs of
a) 42°F or higher?
b) 15°F or lower?
c) between 29°F and 40°F?
change temp to deviations.
a) 2 sigma or greater than mean
b) .7sigma or lower than mean
c) between .7sigma above and 1.8sigma above.
Now use your tables.
To find the probabilities for the given temperature ranges, we can use the standard normal distribution. However, we need to first convert the given values into z-scores using the formula:
z = (x - μ) / σ
where:
- x is the value we want to convert to a z-score,
- μ is the mean of the distribution (22°F in this case),
- σ is the standard deviation of the distribution (10°F in this case).
Let's calculate the z-scores and then use them to find the probabilities for each temperature range:
a) To find the probability of encountering daytime highs of 42°F or higher:
First, calculate the z-score for 42°F:
z = (42 - 22) / 10 = 2
Next, we can find the probability corresponding to a z-value of 2 using a standard normal distribution table or a calculator. Looking up the value for 2 in the standard normal distribution table, we find it to be approximately 0.9772. This means that there's a 0.9772 probability of encountering daytime highs of 42°F or higher.
b) To find the probability of encountering daytime highs of 15°F or lower:
First, calculate the z-score for 15°F:
z = (15 - 22) / 10 = -0.7
Looking up the z-value of -0.7 in the standard normal distribution table, we find its corresponding probability to be approximately 0.2420. Therefore, there's a 0.2420 probability of encountering daytime highs of 15°F or lower.
c) To find the probability of encountering daytime highs between 29°F and 40°F:
First, calculate the z-scores for both temperatures:
For 29°F:
z1 = (29 - 22) / 10 = 0.7
For 40°F:
z2 = (40 - 22) / 10 = 1.8
Next, we need to find the area between these two z-scores. We can subtract the cumulative probability corresponding to the lower z-value from the cumulative probability corresponding to the higher z-value:
p(29°F < x < 40°F) = p(x < 40°F) - p(x < 29°F)
Looking up the z-value of 1.8 in the standard normal distribution table, we find its cumulative probability to be approximately 0.9641. Similarly, the cumulative probability for 0.7 is approximately 0.7580.
Therefore,
p(29°F < x < 40°F) = 0.9641 - 0.7580 = 0.2061
So, there's a 0.2061 probability of encountering daytime highs between 29°F and 40°F.