QUESTION-
machines are used to pack sugar into packets supposedly
containing 1.20kg each.on testing a large number of packets over a
long perodof time ,it was found that the mean weight of the packets
was 1.24kg and the standard deviation was 0.04kg. a particular machine
is selected to check the total weight of each of the 25 packets filled
consecutively by the machine .calculate the limits within which the
weight of the packets should lie assuming that the machine is not been
classified as faulty.
Mean weight of the packets = 1.24 kg
Standard Deviation, SD = 0.04kg
Variance = 0.04^2 = 0.0016
Standard Error, SE = 0.04/sqrt (25) • = 0.04/5 = 0.008
Considering 99.7% confidence level the means will lie between (1.2+3SE) and (1.2-3SE)
Upper limit is 1.224kg
Lower Limit is 1.176kg
To calculate the limits within which the weight of the packets should lie, we need to use the concept of confidence intervals. Confidence intervals provide a range of values within which the true population parameter (in this case, the mean weight of the packets) is likely to fall.
In this scenario, the mean weight of the packets is 1.24kg, and the standard deviation is 0.04kg. Since we don't know the population standard deviation and have a sample size of 25, we can use the t-distribution to calculate the confidence interval.
Now, let's calculate the confidence interval:
1. Determine the level of confidence you want to use. This could be 95% or 99% depending on the required certainty. Let's assume we want a 95% level of confidence.
2. Find the critical value associated with the desired level of confidence. For a sample size of 25 and a 95% level of confidence, the critical value can be found using a t-table or a statistical software. For simplicity, let's assume the critical value is 2.064.
3. Calculate the margin of error using the formula:
Margin of Error = Critical Value * (Standard Deviation / Square Root of Sample Size)
Margin of Error = 2.064 * (0.04 / √25)
Margin of Error = 2.064 * (0.04 / 5)
Margin of Error = 0.01652
4. Calculate the lower limit and upper limit of the confidence interval.
Lower Limit = Mean - Margin of Error
Lower Limit = 1.24 - 0.01652
Lower Limit = 1.22348
Upper Limit = Mean + Margin of Error
Upper Limit = 1.24 + 0.01652
Upper Limit = 1.25652
Therefore, the weight of the packets should lie within the range of 1.22348kg and 1.25652kg, assuming that the machine is not classified as faulty, with a 95% level of confidence.