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 ounce. Is there enough evidence to support the consumer's conjecture at alpha = 0.05? Second, calculate the critical value and degrees of freedom.

Yes there is enough evidence to support----

T = (16-15.6)/(.3/sqrt(8))
df = n-1
df = 8-1 = 7

critical t value = 1.895

Yes there is enough evidence to support----

T = (16-15.6)/(.3/sqrt(8))
df = n-1
df = 8-1 = 7

critical t value = 1.895

Well, well, well, looks like we have some coffee drama here! Let's see if there's enough evidence to support the consumer's suspicions.

To determine whether there's enough evidence, we need to conduct a hypothesis test. In this case, the null hypothesis (H0) would be that the machine is actually filling the jars with exactly 16 ounces of coffee, while the alternate hypothesis (H1) would be that it's not.

To figure out if H0 can be rejected, we can use a t-test. The formula for the t-statistic is:
t = (sample mean - population mean) / (sample standard deviation / √sample size)

Plugging in the values we have, we get:
t = (15.6 - 16) / (0.3 / √8) ≈ -2.37

Now, it's time to consult our t-distribution table to find the critical value for an alpha level of 0.05, considering the degrees of freedom. The degrees of freedom can be calculated as n - 1, where n is the sample size.

In this case, n = 8, so degrees of freedom = 8 - 1 = 7.

So, now we need to find the critical value for alpha = 0.05 and 7 degrees of freedom. According to my calculations, that value is approximately ±2.365.

Since our t-statistic (-2.37) falls outside the range of our critical value (±2.365), we can conclude that there is enough evidence to support the consumer's conjecture. It suggests that the machine is indeed not filling the jars completely with 16 ounces of coffee.

Looks like those coffee jars are getting shortchanged! Time to sip some justice!

To determine if there is enough evidence to support the consumer's conjecture, we can perform a hypothesis test.

Step 1: State the hypotheses.
The null hypothesis (H0): The machine is filling the jars completely (μ = 16 ounces).
The alternative hypothesis (Ha): The machine is not filling the jars completely (μ ≠ 16 ounces).

Step 2: Set the significance level (alpha).
The significance level (alpha) is given as 0.05.

Step 3: Calculate the test statistic.
The test statistic for this hypothesis test is the t-statistic, since we have a small sample size (n < 30) and do not know the population standard deviation. The formula for the t-statistic is:

t = (x̄ - μ) / (s / √n)

where x̄ is the sample mean, μ is the population (target) mean, s is the sample standard deviation, and n is the sample size.

Using the given values:
x̄ = 15.6 ounces (sample mean)
μ = 16 ounces (population mean)
s = 0.3 ounce (sample standard deviation)
n = 8 (sample size)

t = (15.6 - 16) / (0.3 / √8) = -1.294

Step 4: Determine the critical value and degrees of freedom.
To determine the critical value and degrees of freedom, we need the t-distribution table. The degrees of freedom (df) for this test is (n - 1) = (8 - 1) = 7.

At the significance level of 0.05 (two-tailed test), the critical value is obtained by dividing alpha by 2 (0.05 / 2 = 0.025) and looking up the corresponding value in the t-distribution table for df = 7.

Critical value: ±2.365

Step 5: Make a decision.
Since the absolute value of the calculated t-value (-1.294) is less than the critical value (2.365), we fail to reject the null hypothesis.

Step 6: State the conclusion.
Based on the sample data, there is not enough evidence to support the consumer's conjecture that the machine is not filling the jars completely. The sample mean of 15.6 ounces is within the acceptable range of the target mean of 16 ounces.

So, to summarize:
- The calculated test statistic (t-value) is -1.294.
- The critical value is ±2.365.
- The degrees of freedom (df) is 7.
- There is not enough evidence to support the consumer's conjecture at alpha = 0.05.

To determine whether there is enough evidence to support the consumer's conjecture, we can perform a hypothesis test. In this case, the null hypothesis (H0) is that the machine is filling the jars completely, while the alternative hypothesis (Ha) is that the machine is not filling the jars completely.

Step 1: Set up hypotheses
H0: The mean filling amount is 16 ounces (μ = 16)
Ha: The mean filling amount is less than 16 ounces (μ < 16)

Step 2: Select the significance level (α)
The significance level, denoted as α, is the probability of rejecting the null hypothesis when it is true. In this case, α = 0.05, which means we are willing to accept a 5% chance of incorrectly rejecting the null hypothesis.

Step 3: Conduct the test and calculate the test statistic
We will use a one-sample t-test because we have a sample mean and want to compare it to a population mean. The formula for the test statistic is:

t = (sample mean - population mean) / (sample standard deviation / sqrt(sample size))

In this case, the sample mean (x̄) is 15.6 ounces, the population mean (μ) is 16 ounces, the sample standard deviation (s) is 0.3 ounce, and the sample size (n) is 8.

t = (15.6 - 16) / (0.3 / sqrt(8))
≈ -2.828

Step 4: Determine the critical value(s) and degrees of freedom
To determine the critical value, we need to find the t-value that corresponds to a significance level of 0.05 and the degrees of freedom.

The degrees of freedom (df) for a one-sample t-test is given by df = n - 1, where n is the sample size. In this case, df = 8 - 1 = 7.

Using a t-table or statistical software, we find that the critical value for a one-tailed test with 7 degrees of freedom and α = 0.05 is approximately -1.895.

Step 5: Make a decision
If the test statistic falls within the rejection region (i.e., less than the critical value), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

In this case, the test statistic (-2.828) is less than the critical value (-1.895) in the left tail. Therefore, we reject the null hypothesis at α = 0.05.

Step 6: Interpret the result
Based on the given sample data, there is enough evidence to support the consumer's conjecture that the machine is not filling the jars completely.

In summary, the critical value for the one-tailed test at α = 0.05 is approximately -1.895, and the degrees of freedom is 7. The test statistic of -2.828 falls within the rejection region, leading to the rejection of the null hypothesis.