the standard number of cookies in a shipment of bags are normally distributed, with a mean of 64 and a standard deviation of 4. What percent of bags of cookies will contain between 60 and 68 cookies?
To solve this problem, we need to standardize the values using the z-score formula:
z = (x - μ) / σ
Where:
x = the value we want to find the probability for (in this case, between 60 and 68)
μ = the mean of the distribution (64)
σ = the standard deviation of the distribution (4)
For the lower end (60 cookies):
z = (60 - 64) / 4 = -1
For the upper end (68 cookies):
z = (68 - 64) / 4 = 1
Now, we can use a z-table or calculator to find the probability of being between these two z-scores. Using a standard normal distribution table, we can find that the area between -1 and 1 is approximately 0.6827. This means that approximately 68.27% of bags of cookies will contain between 60 and 68 cookies.
are you sure thats correct
I apologize for my mistake. The correct approach to solving this problem is:
First, we can calculate the z-scores for the lower and upper limits of the desired range:
z1 = (60 - 64) / 4 = -1
z2 = (68 - 64) / 4 = 1
Next, we can look up the area under the standard normal distribution curve between these two z-scores using a z-table or calculator. The area between -1 and 1 is approximately 0.6827, as I previously stated.
Finally, we can multiply this area by 100% to obtain the percentage of bags of cookies that will contain between 60 and 68 cookies:
0.6827 x 100% = 68.27%
Therefore, approximately 68.27% of bags of cookies will contain between 60 and 68 cookies. I apologize for any confusion caused by my earlier message.
are you sure
Yes, I am sure that the approach and answer are correct. If you have any further questions or concerns, please let me know.
To calculate the percentage of bags of cookies that will contain between 60 and 68 cookies, we need to calculate the area under the normal distribution curve between these two values.
Step 1: Calculate the z-scores for both 60 and 68 using the formula:
z = (x - μ) / σ
where x is the given value, μ is the mean, and σ is the standard deviation.
For x = 60:
z1 = (60 - 64) / 4 = -1
For x = 68:
z2 = (68 - 64) / 4 = 1
Step 2: Look up the z-scores in the z-table or use a calculator to find the area under the normal distribution curve corresponding to each z-score.
The area for z = -1 is 0.1587 (can be rounded to 0.16) for the left tail of the normal distribution curve.
The area for z = 1 is 0.8413 (can be rounded to 0.84) for the right tail of the normal distribution curve.
Step 3: Subtract the left tail area from the right tail area to find the area between 60 and 68.
Area between 60 and 68 = 0.84 - 0.16 = 0.68
Step 4: Convert the area to a percentage by multiplying by 100.
Percentage of bags of cookies that will contain between 60 and 68 cookies = 0.68 * 100 = 68%
Therefore, approximately 68% of bags of cookies will contain between 60 and 68 cookies.
To find the percentage of bags of cookies that will contain between 60 and 68 cookies, we need to calculate the area under the normal distribution curve between those two values.
Step 1: Convert the values to z-scores
The z-score formula is: z = (x - μ) / σ
For x = 60:
z1 = (60 - 64) / 4 = -1
For x = 68:
z2 = (68 - 64) / 4 = 1
Step 2: Find the area under the curve between the z-scores
To find this area, we need to use a Z-table or calculator. The area under the curve corresponds to the percentage of bags of cookies that fall within the given range.
Using a Z-table, we can look up the values for -1 and 1, and find their corresponding areas.
The Z-table provides the cumulative probability (area under the curve) to the left of a certain z-score. We need to find the difference in probabilities between the values at z1 and z2 to get the area between them.
Let's consult the Z-table:
For z1 = -1, the area to the left is 0.1587.
For z2 = 1, the area to the left is 0.8413.
Step 3: Calculate the area between the z-scores
To find the area between the two z-scores, we subtract the smaller area (z1) from the larger area (z2):
Area = 0.8413 - 0.1587 = 0.6826
Step 4: Convert the area to a percentage
The calculated area represents the percentage of bags of cookies that fall between 60 and 68 cookies. Multiplying it by 100 will give us the percentage:
Percentage = 0.6826 * 100 = 68.26%
Therefore, approximately 68.26% of bags of cookies will contain between 60 and 68 cookies.