Machines are used to pack sugar into packets supposedly containing 1.20 kg each. On testing a large number of packets over a long period of time, it was found that the mean weight of the packets was 1.24 kg and the standard deviation was 0.04 Kg. 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.
MUJHE NAHI PATA.....................
14. If you live in California, the decision to buy earthquake insurance is an important one. A survey revealed that only 133 of 337 randomly selected residences in one California county were protected by earthquake insurance. Calculate the appropriate test statistic to test the hypotheses that at least 40% buy the insurance. (rounded)
A. 0.20
B. 0.39
C. -0.13
D. -0.20
E. -0.39
To calculate the limits within which the weight of the packets should lie, we can use the concept of the confidence interval. The confidence interval provides a range of values within which the true population mean is likely to fall.
Here's how to calculate it:
Step 1: Calculate the sample mean (x̄)
Given that the mean weight of the packets is 1.24 kg.
Step 2: Calculate the standard error (SE)
Standard error measures the variability of the sample mean. It is calculated by dividing the standard deviation by the square root of the sample size. In this case, the sample size is 25 (as 25 packets were checked consecutively).
SE = standard deviation / √sample size
SE = 0.04 kg / √25
SE = 0.04 kg / 5
SE = 0.008 kg
Step 3: Determine the confidence level (C)
The confidence level represents the level of certainty we want to have in our estimate. A common choice is 95%.
Step 4: Determine the critical value (Z)
The critical value is based on the desired confidence level. For a 95% confidence level, the critical value (Z) is approximately 1.96.
Step 5: Calculate the margin of error (ME)
The margin of error is the product of the critical value and the standard error.
ME = critical value * SE
ME = 1.96 * 0.008 kg
ME = 0.01568 kg
Step 6: Calculate the upper and lower limits of the confidence interval
The upper limit is calculated by adding the margin of error to the sample mean, and the lower limit is calculated by subtracting the margin of error from the sample mean.
Upper limit = x̄ + ME
Lower limit = x̄ - ME
Let's calculate it:
Upper limit = 1.24 kg + 0.01568 kg
Upper limit = 1.25568 kg
Lower limit = 1.24 kg - 0.01568 kg
Lower limit = 1.22432 kg
Therefore, the weight of the packets should lie within the range of 1.22432 kg to 1.25568 kg, assuming the machine is not classified as faulty.