The weight of bags of pretzels are normally distributed with a mean of 150 grams and a standard deviation of 5 grams. Bags in the upper 4.5% are too heavy and must be repackaged. Also, the bags in the lower 5% do not meet the minimum weight requirement and must be repackaged. What is the range of weight for a pretzel bag that does not need to be repackaged?
I can't figure out how to set up this problem, I thought this might have been a hypothesis test problem and I drew my bell curve, but now I'm stuck, please help!
Z = (score-mean)/SD
Look in the back of your statistics textbook for a table called something like “area under normal distribution” to find the proportions/probabilities (.045 and .05) and their Z scores. Insert data into equation above to solve for scores.
To solve this problem, you don't need a hypothesis test. Instead, you can use the concept of z-scores.
First, let's find the z-scores corresponding to the upper 4.5% and the lower 5% of the distribution.
For the upper 4.5%, since the weight is considered "too heavy," you need to find the z-score that corresponds to the cumulative probability of 0.955. You can use a standard normal distribution table or a calculator to find this value. The z-score corresponding to a cumulative probability of 0.955 is approximately 1.645.
Now, let's find the z-score for the lower 5%. Since the weight is considered "too light," you need to find the z-score that corresponds to the cumulative probability of 0.05. Using the same method as before, you will find that the z-score corresponding to a cumulative probability of 0.05 is approximately -1.645.
Next, you can use the z-scores to find the actual weights for the bags that do not need to be repackaged.
To find the upper range of weight, you can use the formula:
Upper Range = Mean + (z-score * standard deviation)
Upper Range = 150 + (1.645 * 5) = 150 + 8.225 = 158.225 grams
So, for the bags that do not need to be repackaged, the weight should be less than 158.225 grams.
Similarly, to find the lower range of weight, you use the formula:
Lower Range = Mean + (z-score * standard deviation)
Lower Range = 150 + (-1.645 * 5) = 150 - 8.225 = 141.775 grams
Therefore, for the bags that do not need to be repackaged, the weight should be greater than 141.775 grams.
In conclusion, the range of weight for a pretzel bag that does not need to be repackaged is between 141.775 grams and 158.225 grams.
To find the range of weight for a pretzel bag that does not need to be repackaged, we need to find the weight values that fall within the acceptable range.
First, let's find the z-scores corresponding to the upper 4.5% and lower 5% tail probabilities. Since the weights are normally distributed, we can use the standard normal distribution table or a calculator to find the z-scores.
The upper 4.5% corresponds to the area under the curve to the right. From the standard normal distribution table, we find the z-score that encloses 0.045 (4.5%) from the entire area under the curve, which is approximately 1.645.
The lower 5% corresponds to the area under the curve to the left. From the standard normal distribution table, we find the z-score that encloses 0.050 (5%) from the entire area under the curve, which is approximately -1.645.
Now, we can convert these z-scores back to raw values using the formula:
x = μ + (z * σ), where x is the raw value, μ is the mean, z is the z-score, and σ is the standard deviation.
For the upper range:
x_upper = μ + (z_upper * σ)
= 150 + (1.645 * 5)
= 150 + 8.225
= 158.225 grams
For the lower range:
x_lower = μ + (z_lower * σ)
= 150 + (-1.645 * 5)
= 150 - 8.225
= 141.775 grams
Therefore, the range of weight for a pretzel bag that does not need to be repackaged is between 141.775 grams and 158.225 grams.