Final answer:
We constructed a 95% confidence interval for the population mean of the soft drink, which resulted in (1.9802, 1.9998) liters. A hypothesis test at a 0.05 level of significance revealed that we can reject the null hypothesis, suggesting that the population mean differs from 2.0 liters.
Step-by-step explanation:
To answer the question regarding whether the proper amount of soft drink has been placed in 2-liter bottles, we can use statistical methods.
Part a: Constructing a 95% Confidence Interval
To construct the 95% confidence interval for the population mean, we will use the sample mean (1.99 liters) and the population standard deviation (0.05 liters) provided by the Weights and Measures Department.
Since the sample size is 100, which is a large sample size, we can use the z-distribution for our calculations.
The formula for a confidence interval is:
CI = µ ± (z*σ/√n)
Where ± represents plus or minus, µ is the sample mean, z is the z-score for our confidence level (1.96 for 95%), σ is the standard deviation, and n is the sample size.
Plugging in the values:
CI = 1.99 ± (1.96*0.05/10)
CI = 1.99 ± (0.0098)
CI = (1.9802, 1.9998)
Part b: Hypothesis Testing at the 0.05 Level of Significance
For the hypothesis test, we have the following:
- H0: µ = 2.0 (The population mean is 2 liters)
- Ha: µ ≠ 2.0 (The population mean is not 2 liters)
We use the z-test for testing the hypothesis since the standard deviation is known and the sample size is large.
The test statistic (z) is calculated as:
z = (x - µ) / (σ/√n)
Again, plugging in the values:
z = (1.99 - 2.00) / (0.05/10)
z = -2
Since the absolute value of the observed z is greater than the critical z value of 1.96 for a two-tailed test at the 0.05 significance level, we reject the null hypothesis and conclude that there is evidence that the population mean is different from 2.0 liters.