A flour mill produces flour in small bags before distributing them to wholesalers. The average weight of each bag is 8 kg with a standard deviation of 0.5 kg. A random sample of 50 bags was taken and found that the average weight is 7.8 kg. Using a significance level of 0.01, test the hypothesis that µ = 8 kg against the alternative where µ ≠ 8 kg. (Then, construct the 99% confidence interval)
Z = (mean1 - mean2)/standard error (SE) of difference between means
SEdiff = √(SEmean1^2 + SEmean2^2)
SEm = SD/√n
If only one SD is provided, you can use just that to determine SEdiff.
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability
Hope this helps.
To test the hypothesis that the population mean µ is equal to 8 kg against the alternative that µ is not equal to 8 kg, we can use a t-test for a single sample.
Step 1: State the null and alternative hypotheses:
Null Hypothesis (H0): µ = 8 kg
Alternative Hypothesis (Ha): µ ≠ 8 kg
Step 2: Set the significance level (α):
The significance level (α) is given as 0.01 in the question.
Step 3: Calculate the test statistic:
We can calculate the test statistic using the formula:
t = (x̄ - µ) / (s / √n)
where:
x̄ = sample mean
µ = hypothesized population mean
s = sample standard deviation
n = sample size
Given values:
x̄ = 7.8 kg (sample mean)
µ = 8 kg (hypothesized population mean)
s = 0.5 kg (standard deviation)
n = 50 (sample size)
Plugging in the values, we get:
t = (7.8 - 8) / (0.5 / √50)
Step 4: Determine the critical value(s):
Since the alternative hypothesis is two-sided (µ ≠ 8 kg), we need to consider the critical values from the t-distribution with (n-1) degrees of freedom. Since the sample size is large (n = 50), we can assume the t-distribution is approximately normal.
Using a significance level of 0.01, the critical value can be found using a t-table or calculator. In this case, the critical value is approximately ±2.685.
Step 5: Make a decision:
If the test statistic falls outside the critical value range, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Step 6: Calculate the p-value:
We can calculate the p-value associated with the test statistic using a t-distribution calculator or using software. The p-value is the probability of obtaining a test statistic as extreme or more extreme than the one observed, assuming the null hypothesis is true.
Step 7: Interpret the result:
If the p-value is less than the significance level (α), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
To construct a 99% confidence interval, we can use the formula:
CI = x̄ ± (t * (s / √n))
where:
x̄ = sample mean
s = sample standard deviation
n = sample size
t = critical value from the t-distribution with (n-1) degrees of freedom
Plugging in the values, we get:
CI = 7.8 ± (2.685 * (0.5 / √50))
Now, we can calculate the actual values for the confidence interval.
To test the hypothesis and construct the confidence interval, we need to follow these steps:
Step 1: State the null and alternative hypothesis.
The null hypothesis (H0) assumes that there is no significant difference between the sample mean and the population mean. In this case, H0: µ = 8 kg.
The alternative hypothesis (Ha) assumes that there is a significant difference between the sample mean and the population mean. In this case, Ha: µ ≠ 8 kg.
Step 2: Determine the test statistic.
Since we know the population standard deviation (0.5 kg) and the sample size (50 bags), we can use the z-test statistic formula:
z = (sample mean - population mean) / (population standard deviation / sqrt(sample size))
Step 3: Set the significance level.
The significance level (α) is given as 0.01.
Step 4: Calculate the test statistic value.
Using the given values in the formula:
z = (7.8 - 8) / (0.5 / sqrt(50))
Step 5: Find the critical value(s).
Since the alternative hypothesis is two-tailed (µ ≠ 8 kg), we need to find the critical values in the z-distribution. Using a significance level of 0.01, we split 0.01 into two tails, resulting in 0.005 in each tail. We can find these critical values using a z-table or a calculator.
Step 6: Compare the test statistic value with the critical value(s).
If the absolute value of the test statistic is greater than the critical value(s), we reject the null hypothesis; otherwise, we fail to reject the null hypothesis.
Step 7: Calculate the confidence interval.
To construct a 99% confidence interval, we need to find the margin of error and subtract/add it to the sample mean. The margin of error (ME) can be calculated as:
ME = critical value * (population standard deviation / sqrt(sample size))
Then, the confidence interval formula becomes:
Confidence interval = sample mean ± margin of error
Step 8: Interpret the results.
Based on the outcome of the hypothesis test and the confidence interval, we can draw conclusions about the population mean.
Note: Since the calculations involve specific values, it's recommended to use a calculator or statistical software to get the exact results.