A Manufacturer of torque converters claim only 1% of them are defective.
A. Calculate the Mean number of defective converters in a shipment of 500 converters.
B. Calculate the standard deviation in the defective torque converters in a shipment of 500 converters.
C. Would it be unusual for 9 converters (out of a shipment of 500 converters) to be defective?
To calculate the mean and standard deviation in this scenario, we can use the properties of a binomial distribution. In a binomial distribution, we have two possible outcomes (defective or not defective), and we know the probability of the defective outcome (1%).
A. To calculate the mean number of defective converters in a shipment of 500 converters, we multiply the total number of converters (500) by the probability of a defective converter (1% or 0.01).
Mean = Number of Converters * Probability of Defective Converter
Mean = 500 * 0.01
Mean = 5
Therefore, the mean number of defective converters in a shipment of 500 converters is 5.
B. To calculate the standard deviation in the defective torque converters in a shipment of 500 converters, we use the formula for the standard deviation of a binomial distribution:
Standard Deviation = sqrt(Number of Converters * Probability of Defective Converter * (1 - Probability of Defective Converter))
Standard Deviation = sqrt(500 * 0.01 * (1 - 0.01))
Standard Deviation = sqrt(500 * 0.01 * 0.99)
Standard Deviation = sqrt(4.95)
Standard Deviation ≈ 2.22
The standard deviation in the defective torque converters in a shipment of 500 converters is approximately 2.22.
C. To determine whether it would be unusual for 9 converters out of a shipment of 500 converters to be defective, we can use the concept of standard deviation. If the number of defective converters is more than 2 standard deviations away from the mean, it would be considered unusual.
In this case, the mean number of defective converters is 5, and the standard deviation is approximately 2.22.
Number of Defective Converters = 9
Z-Score = (Number of Defective Converters - Mean) / Standard Deviation
Z-Score = (9 - 5) / 2.22
Z-Score ≈ 1.8
By looking up the area under the normal distribution curve corresponding to a Z-Score of 1.8, we can determine the probability of getting 9 or more defective converters. This probability indicates whether it would be unusual or not.
However, without knowing the specific cutoff for "unusual," we cannot make a definitive determination. Typically, a Z-Score greater than 2 or less than -2 is considered unusual, which corresponds to probabilities less than 5%. In this case, the Z-Score of 1.8 falls between 1 and 2 standard deviations, making it less likely but not necessarily unusual.