In Exercises 8 and 9, assume that women have standing eye heig...

Question

In Exercises 8 and 9, assume that women have standing eye heights that are normally distributed with a mean of 59.7 in. and a standard deviation of 2.5 in. (based on anthropometric survey data from Gordon, Churchill, et al.).

Significance Instead of using 0.05 for identifying significant values, use the criteria that a value x is significantly high if P(x or greater) ≤ 0.01 and a value is significantly low if P(x or less) ≤ 0.01. Find the standing eye heights of women that separate significant values from those that are not significant. Using these criteria, is a woman’s standing eye height of 67 in. significantly high?

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says suppose the weight of newborn babies in a hospital are normally distributed with a mean of 7.4 pounds and a standard deviation of 1.1 pounds. Using the criteria that a value X is significantly high if the probability of X or greater is less than or equal to 0.01 and significantly low if the probability of X or less is less than or equal to 0.01. Determine the weight. Separate significant values from those that are not significant. Is a weight of 10 pounds significantly high? We have two possible choices as our answers. For choice A, we have yes. 10 pounds is significantly high, and for choice B, we have no. 10 pounds is not significantly high. Now, in the problem, we're given the criteria for significantly high values and significantly low values. So if we look at the condition, or significantly high values, We have the probability of X is greater than equal to lowercase x. Which is going to be less than or equal to 0.01. And for significantly low values. We have the probability of capital X is less than or equal to lowercase x. And that probability has to be less than or equal to 0.01. Now, based on these probabilities, we see that these significantly high values. Correspond to the 99th percentile and significantly low values correspond to the 1 percentile. So we can actually find the Z scores that corresponds to those percentiles for significantly high percentiles. We look at our tables, we'll see that P of Z is greater than or equal to 2.33, and that is approximately equal to 0.01, so that's gonna be the 99th percentile. And so that corresponds to a Z value, that is equal to 2.33 for that limit. Now, for the significantly low case, if we look for the 1 percentile, we'll see that P of Z is less than or equal to minus 2.33. And that probability is approximately equal to 0.01. And so that corresponds to a Z value that is equal to minus 2.33, or that limit for the lower significance. So, what we can do now is convert these Z scores into actual weights. And so we're called to convert a Z score to an actual value, that we're going to have the formula. X is equal to mu. Plus Z multiplied by sigma. Where mu here is our mean and sigma is our standard deviation, and we were given both of those values in the problem. So, if you look at the highly significant case, Then We will have X is equal to. And from you, we will have the 7.4 pounds, we'll have 7.4. Plus, for a Z score, we have the 2.33, and then that gets multiplied by our standard deviation of 1.1 pounds, so multiply that by 1.1. And when we calculate this value, this is going to be equal to. 9.96. So that means for a value to be significantly high. It has to be greater than or equal to 9.96 pounds. So that's one of our limits that we were looking for. That separate the significant values from the non-significant ones. Next, we'll look at the significantly low case. And we'll calculate our value for X, and that's going to be equal to 7.4, since we have the same mean. Plus, here we'll use our new Z value of minus 2.33. That gets multiplied by our standard deviation of 1.1. And evaluating this expression, this is going to be equal to 4.84. That means for a value to be significantly low. It has to be less than or equal to. 4.84 pounds. And so that is the other value that we were looking for that separates. Highly significant value or significant values from non-significant values. So now we just need to answer our question, is a weight of 10 pounds significantly high? And if you look at 10 pounds, that's going to be greater than our 9.96 pounds, which is going to be the lower limit for a value to be significantly high. And so, since the 10 pounds is greater than the 9.96 pounds, that means that this value is significantly high. So the answer for this question is yes. And if we look at our answer choices, that corresponds to answer choice A. So I hope that this explanation has been useful, and I'll see you in the next video.

Key Concepts

Normal Distribution

Z-Score

Significance Level

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