Please answer with full solution! Thanks for the help

Question

Readings on a thermometer are normally distributed with a mean of 0C and a standard deviation of 1.00.

What is the probability that a randomly selected

thermometer reads:

less than 1.772°C.

greater than -2.016°C.

greater than 1.501°C.

between 1.178°C and 2.668°C.

Answer 1

Hi Stacy,

What a great introductory statistics question!

Basically, you are asking what probabilities are associated with different z-scores. As you may know, Z-scores are standardized values that allow you to turn any normal distribution into a standardized normal distribution with mean zero (μ=0.0) and standard deviation of one (σ=1.0). To calculate the z-score of any observation, simply subtract the mean and divide by the standard deviation. If we knew people had average heights of 5'8" and standard deviations of 6", then a height of 6" would result in a z-score of ((6'1" - 5'8)/6") = 5"/6" = .83 In this thermometer case, this calculation is trivial because the observations already have a mean of zero and a standard deviation of one. The measurements are already z-scores.

Once you understand how to calculate z-scores, you can connect z-scores to a cumulative probability distributions. The cumulative probability distribution is related to the calculus integral and tells you what percentage of observations lie between two values. For this, you will need a z-table or function call that calculates cumulative normal values. Here are some common examples: 2.5% of observations have a z-score less than -1.96 or greater than +1.96, meaning about 95% of observations lie within about two standard deviations of the mean (1.96 is about 2, 2.5% x 2 = 5%). Similarly, about two-thirds (68%) of observations lie within about one standard deviation. Half of observations have z-scores greater than or less 0.0.

So to answer these problems, look at your z-table, find your z-score (e.g. 1.772) with the resulting cumulative probability in the left or right tail and decide what percentage of observations lie to the left of 1.772 and what percentage of observations lie to the right of 1.772. I recommend you draw out a standard normal curve and write in what percentage of observations lie in which regions and check that all the regions add up to 100% before reporting the percentage for the region you care about (e.g. less than 1.772).

If you want to go over specific problems or learn how these concepts apply to your personal interests or other critical real-world phenomena, just send me a message!