Math  /  Data & Statistics

QuestionAt a water bottling factory, a machine is supposed to put 2 liters of water into the bottles. After an overhaul, management thinks the machine is no longer putting the correct amount of water in. They sample 20 bottles and and find an avg of 2.10 L of water with standard deviation of 0.33 L . Test the claim at 0.01 level of significance.

Studdy Solution

STEP 1

1. The null hypothesis is that the machine is putting exactly 2 liters of water into the bottles.
2. The alternative hypothesis is that the machine is not putting 2 liters of water into the bottles.
3. The sample size is 20 bottles.
4. The sample mean is 2.10 liters.
5. The sample standard deviation is 0.33 liters.
6. The level of significance is 0.01.

STEP 2

1. Formulate the hypotheses.
2. Calculate the test statistic.
3. Determine the critical value(s).
4. Make a decision to accept or reject the null hypothesis.

STEP 3

Formulate the hypotheses:
- Null hypothesis (H0H_0): μ=2\mu = 2 liters - Alternative hypothesis (HaH_a): μ2\mu \neq 2 liters

STEP 4

Calculate the test statistic using the formula for the t-statistic:
t=xˉμs/nt = \frac{\bar{x} - \mu}{s/\sqrt{n}}
where xˉ=2.10\bar{x} = 2.10, μ=2\mu = 2, s=0.33s = 0.33, and n=20n = 20.
t=2.1020.33/20=0.100.07381.355t = \frac{2.10 - 2}{0.33/\sqrt{20}} = \frac{0.10}{0.0738} \approx 1.355

STEP 5

Determine the critical value(s) for a two-tailed test at the 0.01 level of significance with n1=19n - 1 = 19 degrees of freedom. Using a t-distribution table, the critical t-value is approximately ±2.861\pm 2.861.

STEP 6

Make a decision:
Since the calculated t-statistic 1.3551.355 is less than the critical value 2.8612.861, we fail to reject the null hypothesis.
Conclusion: There is not enough evidence at the 0.01 level of significance to conclude that the machine is putting a different amount of water than 2 liters.

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord