A machine is designed to fill jars with 16 ounces of coffee. A consumer suspects that the machine is not filling the jars completely. A sample of 8 jars has a mean of 15.6 ounces and a standard deviation of 0.3 ounces. Is there enough evidence to support the consumer's claim at a=0.10?
Z = (score-mean)/SEm
SEm = SD/√n
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability for the Z score.
To determine whether there is enough evidence to support the consumer's claim, we can conduct a hypothesis test using the given information.
Step 1: State the hypotheses.
- Null hypothesis (H₀): The machine is filling the jars with 16 ounces of coffee.
- Alternative hypothesis (H₁): The machine is not filling the jars with 16 ounces of coffee.
Step 2: Set the significance level (α).
The given α level is 0.10, which means we are willing to accept a 10% chance of making a Type I error (rejecting the null hypothesis when it is actually true).
Step 3: Calculate the test statistic.
In this case, we will use a one-sample t-test because we are comparing the sample mean to a known population mean (16 ounces). The formula for the t-score is:
t = (sample mean - population mean) / (sample standard deviation / sqrt(sample size))
Sample mean: 15.6 ounces
Population mean: 16 ounces
Sample standard deviation: 0.3 ounces
Sample size (n): 8 jars
Plugging in the values:
t = (15.6 - 16) / (0.3 / sqrt(8))
Step 4: Determine the critical value.
Since our alternative hypothesis is "not equal to" (two-tailed test), we need to find the critical t-value for a two-tailed test with a 0.10 significance level and degrees of freedom (df) equal to the sample size minus 1 (n - 1). Look up the critical t-value in the t-distribution table or use a statistical calculator.
Step 5: Compare the test statistic with the critical value.
If the absolute value of the test statistic is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Step 6: Make a conclusion.
If we reject the null hypothesis, there is enough evidence to support the consumer's claim. If we fail to reject the null hypothesis, there is not enough evidence to support the consumer's claim.
By completing these steps, you can determine whether there is enough evidence to support the consumer's claim.