if 5% of all calculators are defective, find the mean, variance, and standard deviation of a lot of 400 calculators. round your final answer to two decimal places if necessary.
To find the mean, variance, and standard deviation of a lot of 400 calculators with a 5% defect rate, we can follow these steps:
Step 1: Calculate the number of defective calculators.
The number of defective calculators can be found by multiplying the total number of calculators (400) by the defect rate (5%).
Number of defective calculators = 400 * 5% = 400 * 0.05 = 20 calculators.
Step 2: Calculate the number of non-defective calculators.
The number of non-defective calculators is the difference between the total number of calculators and the number of defective calculators.
Number of non-defective calculators = Total number of calculators - Number of defective calculators = 400 - 20 = 380 calculators.
Step 3: Calculate the mean (average).
The mean can be calculated by summing the number of defective calculators weighted by 1 and the number of non-defective calculators weighted by 0.
Mean = (Number of defective calculators * 1 + Number of non-defective calculators * 0) / Total number of calculators
Mean = (20 * 1 + 380 * 0) / 400
Mean = 20 / 400
Mean = 0.05
Step 4: Calculate the variance.
The variance can be calculated by multiplying the defect rate (5%) by the probability of a calculator being non-defective (95%) and then multiplying the result by the total number of calculators (400).
Variance = Defect rate * (1 - Defect rate) * Total number of calculators
Variance = 0.05 * 0.95 * 400
Variance = 0.0475 * 400
Variance = 19
Step 5: Calculate the standard deviation.
The standard deviation can be calculated by taking the square root of the variance.
Standard deviation = √Variance
Standard deviation = √19
Rounding to two decimal places, the mean is 0.05, the variance is 19, and the standard deviation is approximately 4.36.
To find the mean, variance, and standard deviation of a lot of 400 calculators, you can follow these steps:
1. Find the number of defective calculators in the lot: Multiply the total number of calculators (400) by the defect rate (5%) to obtain the number of defective calculators.
Number of defective calculators = 400 * 0.05 = 20
2. Find the number of non-defective calculators: Subtract the number of defective calculators from the total number of calculators.
Number of non-defective calculators = 400 - 20 = 380
3. Calculate the mean (μ): The mean represents the average value.
Mean = (Number of defective calculators * value of defective calculators) + (Number of non-defective calculators * value of non-defective calculators) / Total number of calculators
In this case, since the value of defective calculators is not given, we assume it to be 0.
Mean = (20 * 0) + (380 * 1) / 400
Mean = 380 / 400
Mean = 0.95
4. Calculate the variance (σ^2): The variance measures the spread or dispersion of the data from the mean.
Variance = ((Number of defective calculators * (value of defective calculators - mean)^2) + (Number of non-defective calculators * (value of non-defective calculators - mean)^2)) / Total number of calculators
In this case, since the value of defective calculators is assumed to be 0, the term involving defective calculators becomes 0.
Variance = (20 * 0^2 + (380 * (1 - 0.95)^2)) / 400
Variance = (380 * (0.05)^2) / 400
Variance = (380 * 0.0025) / 400
Variance = 0.002375
5. Calculate the standard deviation (σ): The standard deviation is the square root of the variance and represents the average amount of variation or dispersion from the mean.
Standard deviation = √Variance
Standard deviation = √0.002375
Standard deviation ≈ 0.0487 (rounded to two decimal places)
Therefore, the mean is 0.95, the variance is 0.002375, and the standard deviation is approximately 0.0487.