question- a packaging device is set to fill detergent powder packets
with a mean weight of 5 kg .the standered deviation is known to be
0.01kg.these are known to drift upward 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 significance.
wscvhy
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 if the mean weight produced by the machine has increased, we can use hypothesis testing with a significance level of 5%.
Step 1: Define the null and alternative hypotheses:
- Null hypothesis (H0): The mean weight produced by the machine has not increased (μ = 5 kg).
- Alternative hypothesis (Ha): The mean weight produced by the machine has increased (μ > 5 kg).
Step 2: Select the appropriate test statistic:
Since the sample size is large (n = 100) and we know the population standard deviation, we can use the Z-test. The formula for the test statistic is:
Z = (sample mean - population mean) / (population standard deviation / sqrt(sample size))
Step 3: Calculate the test statistic:
Given:
Sample mean (x̄) = 5.03 kg
Population mean (μ) = 5 kg
Population standard deviation (σ) = 0.01 kg
Sample size (n) = 100
Z = (5.03 - 5) / (0.01 / sqrt(100))
Z = 0.03 / (0.01 / 10)
Z = 3
Step 4: Determine the critical value:
Since we are conducting a one-tailed test with a significance level of 5%, the critical value is found using a Z-table or calculator. At a 5% significance level, the critical Z-value is approximately 1.645.
Step 5: Compare the test statistic with the critical value:
Since the calculated Z-value (3) is greater than the critical Z-value (1.645), we reject the null hypothesis.
Step 6: Draw a conclusion:
Based on the sample data, we can conclude that there is evidence to suggest that the mean weight produced by the machine has increased.
Note: The 5% level of significance means that we are willing to accept a 5% chance of making a Type I error (rejecting the null hypothesis when it is true).
To determine if the mean weight produced by the machine has increased, we can perform a hypothesis test using the given sample data.
Here are the steps to conduct the hypothesis test:
Step 1: State the hypotheses
Null Hypothesis (H₀): The mean weight produced by the machine has not increased.
Alternative Hypothesis (H₁): The mean weight produced by the machine has increased.
Step 2: Set the significance level
The significance level (α) is given as 5%, which means we will reject the null hypothesis if the probability of obtaining the sample mean is less than 5%.
Step 3: Compute the test statistic
We will use the t-test because the population standard deviation is unknown. The formula for the t-test statistic is:
t = (sample mean - hypothesized mean) / (sample standard deviation / sqrt(sample size))
Given:
Sample mean (x̄) = 5.03 kg
Hypothesized mean (μ₀) = 5 kg
Sample standard deviation (s) = 0.21 kg
Sample size (n) = 100
t = (5.03 - 5) / (0.21 / sqrt(100))
Step 4: Determine the critical value
Since the alternative hypothesis is one-tailed (we are testing for an increase only), we need to find the critical t-value for a one-tailed test with 99 degrees of freedom at the 5% level of significance.
Using a t-table or statistical software, the critical t-value for a one-tailed test at the 5% level of significance is approximately 1.660.
Step 5: Make the decision
If the computed t-value is greater than the critical t-value, we reject the null hypothesis; otherwise, we fail to reject the null hypothesis.
Compare the computed t-value with the critical t-value:
If t > tₐ, reject the null hypothesis.
If t ≤ tₐ, fail to reject the null hypothesis.
Step 6: Calculate the p-value
The p-value is the probability of obtaining a test statistic as extreme as (or more extreme than) the observed value, assuming the null hypothesis is true. If the p-value is less than the significance level (α), we reject the null hypothesis.
Using statistical software or a t-distribution calculator, we can calculate the p-value.
Step 7: Make the conclusion
If the p-value is less than the significance level (α), we reject the null hypothesis and conclude that the mean weight produced by the machine has increased. Otherwise, we fail to reject the null hypothesis.
Note: Since we don't have the actual data values, I am unable to provide the exact values for the test statistic and p-value. You will need to perform the calculations using the given data.