A packaging device is set to fill detergent power packets with a mean weight of 5 Kg. The standard deviation is known to be 0.01 Kg. These are known to drift upwards over a period of time due to machine fault, which is not tolerable. A random sample of 100 packets is taken and weighed. This sample has a mean weight of 5.03 Kg and a standard deviation of 0.21 Kg. Can we calculate that the mean weight produced by the machine has increased? Use 5% level of significance.
To determine whether the mean weight produced by the machine has increased, we can perform a hypothesis test.
Step 1: State the null and alternative hypotheses.
The null hypothesis (H0) assumes that there is no significant increase in the mean weight produced by the machine.
H0: μ = 5 (mean weight is 5 Kg)
The alternative hypothesis (Ha) assumes that there is a significant increase in the mean weight produced by the machine.
Ha: μ > 5 (mean weight is greater than 5 Kg)
Step 2: Choose the appropriate test statistic.
Since we have a sample mean and know the population standard deviation, we can use the Z-test.
Step 3: Determine the rejection region.
Since we are conducting a one-tailed test (the alternative hypothesis is that the mean weight is greater than 5 Kg), we need to find the critical value corresponding to the chosen significance level (5%). This can be found using a Z-table or a calculator.
For a 5% significance level, the critical value is approximately 1.645.
Step 4: Calculate the test statistic.
The test statistic for a Z-test is calculated by subtracting the hypothesized mean from the sample mean, divided by the standard deviation divided by the square root of the sample size.
Z = (sample mean - hypothesized mean) / (population standard deviation / sqrt(sample size))
Z = (5.03 - 5) / (0.01 / sqrt(100))
Z = 0.03 / 0.01
Z = 3
Step 5: Compare the test statistic with the critical value.
If the test statistic falls in the rejection region (i.e. it is greater than the critical value), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
In this case, the test statistic (Z = 3) is greater than the critical value (1.645), so we reject the null hypothesis.
Step 6: Make the Conclusion.
Based on the results of our hypothesis test, with a 5% significance level, we can conclude that there is sufficient evidence to suggest that the mean weight produced by the machine has increased.