Please help me, it would be greatly appreciated.
1.) A set of data was collected as shown in the table.
Data value: Frequency:
5. 9
6. 9
7. 14
8. 9
9. 13
10. 7
11. 9
The standard deviation of the data is
A..) 1.90
B.) 2.00
C.) 2.33
D.) 7.93
2.) A group of apples from a particular grower were found to have diameters with a mean of 68.0mm and a standard deviation of 3.9mm. It is expected that 75% of apples will have diameters larger than
A.) 65.4mm
B.) 70.9mm
C.) 71.9mm
D.) 70.6mm
3.) A group of apples from a particular grower were found to have diameters with a mean of 67.0mm and a standard deviation of 4.1mm. It is expected that 30% of apples will have diameters larger than
A.) 71.1mm
B.) 64.9mm
C.) 68.2mm
D.) 69.1mm
To find the standard deviation of a set of data, you can use the following formula:
Standard Deviation = sqrt(sum((x - mean)^2) / (N - 1))
where:
- x represents each data value,
- mean is the average of the data values,
- sum() represents the sum of all the values inside the parentheses,
- N is the total number of data values.
Let's calculate the standard deviation for the first question:
1.) A set of data was collected as shown in the table.
Data value: Frequency:
5. 9
6. 9
7. 14
8. 9
9. 13
10. 7
11. 9
First, we need to find the mean of the data set. To do this, multiply each data value by its corresponding frequency and sum the results. Then, divide the sum by the total number of frequencies. In this case, the total number of frequencies is 9+9+14+9+13+7+9 = 70.
Mean = (5*9 + 6*9 + 7*14 + 8*9 + 9*13 + 10*7 + 11*9) / 70
Now, let's calculate the sum of the squared differences from the mean. Subtract the mean from each data value, square the result, multiply it by its frequency, and sum all the results.
Sum of squared differences from the mean = (9 * (5 - mean)^2) + (9 * (6 - mean)^2) + (14 * (7 - mean)^2) + (9 * (8 - mean)^2) + (13 * (9 - mean)^2) + (7 * (10 - mean)^2) + (9 * (11 - mean)^2)
Finally, divide the sum of squared differences from the mean by (N - 1) and take the square root to find the standard deviation.
Standard Deviation = sqrt(Sum of squared differences from the mean / (N - 1))
Now you can calculate the standard deviation and choose the correct answer.
For the second and third questions, you can use the Z-score formula to find the value that corresponds to a given percentage. The formula is as follows:
Z = (x - mean) / standard deviation
where:
- x is the value you want to find (diameter in this case),
- mean is the mean of the data,
- standard deviation is the standard deviation of the data.
To find the value corresponding to a certain percentage, you can rearrange the formula as:
x = mean + Z * standard deviation
Now you can substitute the given values into the formula to find the diameter for each question. Calculate the Z-score using the given percentages and the mean and standard deviation provided. Then, substitute the Z-score into the formula to find the diameter.
I hope this helps! If you have any further questions, please let me know.
1.) To find the standard deviation of the data, you can follow these steps:
Step 1: Calculate the mean of the data:
To find the mean, you need to multiply each data value by its corresponding frequency, then sum up these products, and finally divide by the total frequency.
Mean = (5 * 9 + 6 * 9 + 7 * 14 + 8 * 9 + 9 * 13 + 10 * 7 + 11 * 9) / (9 + 9 + 14 + 9 + 13 + 7 + 9)
Step 2: Calculate the squared deviation for each data value:
To find the squared deviation, subtract the mean from each data value, and then square the result.
Squared Deviation = (5 - Mean)^2 * Frequency + (6 - Mean)^2 * Frequency + (7 - Mean)^2 * Frequency + (8 - Mean)^2 * Frequency + (9 - Mean)^2 * Frequency + (10 - Mean)^2 * Frequency + (11 - Mean)^2 * Frequency
Step 3: Calculate the variance:
To find the variance, sum up all the squared deviations, then divide by the total frequency.
Variance = (Squared Deviation for 5 + Squared Deviation for 6 + Squared Deviation for 7 + Squared Deviation for 8 + Squared Deviation for 9 + Squared Deviation for 10 + Squared Deviation for 11) / (9 + 9 + 14 + 9 + 13 + 7 + 9)
Step 4: Calculate the standard deviation:
To find the standard deviation, take the square root of the variance.
Standard Deviation = sqrt(Variance)
Now you can perform these calculations to find the answer.
2.) To find the diameter larger than a certain value, you can calculate the z-score and refer to the standard normal distribution table.
Step 1: Calculate the z-score:
To find the z-score, subtract the mean from the given diameter value, then divide by the standard deviation.
z-score = (Given Diameter - Mean) / Standard Deviation
Step 2: Use the z-score table:
Consult the standard normal distribution table or use a calculator with a built-in z-score calculator to find the corresponding percentile for the z-score.
Step 3: Subtract the percentile from 1:
Since we're interested in the percentage larger than the given diameter value, subtract the percentile found from 1. This will give you the percentage of observations larger than the given diameter.
Now, you can calculate the diameter values for both questions 2 and 3 using these steps.
#1 See the excellent step-by-step article at
https://mathway.com/examples/Statistics/Frequency-Distribution/Finding-the-Standard-Deviation?id=1040
Use your data values where they used interval-midpoint values (M).
For the others, use the handy tool found at
http://davidmlane.com/hyperstat/z_table.html