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?
The annual Salary of an electrical engineer is given in terms of the years of experience by the table below. Find the equation of linear regression for the above data and obtain the expected salary for an engineer with 48 years of experience. Round to the nearest $100.
The beauty of this webpage is that you don't even have to find the z-scores, but you can if you want.
If works perfectly for your problem.
http://davidmlane.com/hyperstat/z_table.html
First, if you have a question, it is much better to put it in as a separate post in <Post a New Question> rather than attaching it to a previous question, where it is more likely to be overlooked.
Second, we do not have access to your table.
To find the probability of encountering specific temperature ranges, we need to use the concept of the standard normal distribution. We will convert the given temperatures to standardized z-scores and then use a standard normal distribution table or a calculator to find the probabilities.
The formula to calculate the z-score is:
z = (x - μ) / σ
Where:
- x is the given value (temperature)
- μ is the mean of the distribution
- σ is the standard deviation of the distribution
Let's calculate the probabilities for each scenario:
a) To find the probability of encountering temperatures 42°F or higher, we need to calculate the z-score for 42°F using the formula above:
z = (42 - 22) / 10 = 2
Now, we need to find the probability corresponding to a z-score of 2. Using a standard normal distribution table or calculator, we can find that the probability associated with a z-score of 2 is approximately 0.9772.
Therefore, the probability of encountering daytime highs of 42°F or higher is approximately 0.9772 or 97.72%.
b) To find the probability of encountering temperatures 15°F or lower, we calculate the z-score for 15°F:
z = (15 - 22) / 10 = -0.7
Using the standard normal distribution table or calculator, we can find that the probability associated with a z-score of -0.7 is approximately 0.2420.
Therefore, the probability of encountering daytime highs of 15°F or lower is approximately 0.2420 or 24.20%.
c) To find the probability of encountering temperatures between 29°F and 40°F, we will calculate the z-scores for both values:
For 29°F:
z1 = (29 - 22) / 10 = 0.7
For 40°F:
z2 = (40 - 22) / 10 = 1.8
Now, we need to find the probability between these two z-scores. Using a standard normal distribution table or calculator, we can find that the probability associated with a z-score of 0.7 is approximately 0.7580, and the probability associated with a z-score of 1.8 is approximately 0.9641.
To find the probability between these two values, we subtract the smaller probability from the larger probability:
0.9641 - 0.7580 = 0.2061
Therefore, the probability of encountering daytime highs between 29°F and 40°F is approximately 0.2061 or 20.61%.
Note: Keep in mind that these calculations assume a normal distribution for the daytime high temperatures in January at the ski area in Vermont.