Weights of the vegetables in a field are normally distributed. From a sample Carl Cornfield determines the mean weight of a box of vegetables to be 180 oz. with a standard deviation of 8 oz. He wonders what percent of the vegetable boxes he has grouped for sale will have a weight between 169 oz. and 191 oz. Carl decides to answer the following questions about the population of vegetables from these sample statistics. You will use some answers more than once. Carl calculates the z-score corresponding to the weight 169 oz. (to the nearest tenth). Using the table, Carl sees the percentage associated with this z-score is
To calculate the z-score, we can use the formula:
z = (x - μ) / σ
where x is the weight of the box, μ is the mean weight, and σ is the standard deviation.
In this case, x = 169 oz, μ = 180 oz, and σ = 8 oz.
Plugging these values into the formula:
z = (169 - 180) / 8 ≈ -1.375
To find the percentage associated with this z-score, we can consult a standard normal distribution table or use a statistical calculator. From the table, the percentage associated with a z-score of approximately -1.375 is about 0.084 or 8.4%.
To calculate the z-score corresponding to the weight of 169 oz, we can use the formula:
z = (x - μ) / σ
where:
- x is the weight (169 oz)
- μ is the mean weight of a box of vegetables (180 oz)
- σ is the standard deviation of the weights (8 oz)
Substituting the given values into the equation:
z = (169 - 180) / 8
z = -11 / 8
z ≈ -1.375
Now, using a standard normal distribution table or a calculator that provides the percentage associated with a given z-score, we can find the answer.
Looking up the z-score of -1.375 in the standard normal distribution table, we can approximate the percentage associated with it. The table provides the cumulative probability (from the left) up to a certain z-score.
The percentage associated with a z-score of -1.375 is approximately 0.084, or 8.4% (rounded to one decimal place).
Therefore, Carl determines that approximately 8.4% of the vegetable boxes he has grouped for sale will have a weight between 169 oz. and 191 oz.
z-score for the 169 = (169-180)/8 = -1.375
x-score for the 191 = (191-180)/8 = +1.375
look up value for 1.375 and the value for -1.375
subtract the two values
You should get .8309
I recommend this webpage for these type of questions.
http://www.wolframalpha.com/input/?i=plot+y+%3D+%28x%5E2%2B4x-5%29%2F%28x-2%29+%2C+y+%3D+x%2B6
The beauty of this page is that it lets you enter the mean and SD directly without even finding the z-score.
Personally, I don't see any difference in going to some table that "somebody" created ages ago, and using this page. In either case you are not actually doing the arithmetic.
notice you can just enter 180 for the mean, 8 for the sd, click on the "between" and enter 169 and 191 to get 0.8309
Bookmark this webpage for future use.