Quality control at a soda bottler suspects that the standard deviation in the mean volume of bottles of soda being filled by the bottler is intolerably high. In a sample of 14 bottles, the sample standard deviation is 1.5 ml. Use this sample to test the null hypothesis Ho: 0 = 1.2 against the alternative hypothesis Hy: 0 > 1.2 at 0.05 significance level.
To test the hypothesis, we can use a one-sample t-test. Here are the steps to calculate and interpret the test:
Step 1: State the null and alternative hypotheses.
Null hypothesis (Ho): The population mean volume of bottles (μ) is equal to 1.2 ml.
Alternative hypothesis (Hy): The population mean volume of bottles (μ) is greater than 1.2 ml.
Step 2: Determine the significance level.
The significance level is given as 0.05.
Step 3: Calculate the test statistic.
The test statistic for a one-sample t-test is calculated using the formula:
t = (x̄ - μ) / (s / √n)
Where:
x̄ = sample mean
μ = hypothesized population mean
s = sample standard deviation
n = sample size
In this case, the sample mean (x̄) is unknown, so we need to estimate it using the null hypothesis. Therefore:
x̄ = 1.2 (null hypothesis value)
μ = 1.2 (null hypothesis value)
s = 1.5 ml (given sample standard deviation)
n = 14 (given sample size)
Substituting the values into the formula, we get:
t = (1.2 - 1.2) / (1.5 / √14)
t = 0 / 0.401
Note: The numerator is zero because the sample mean is equal to the hypothesized mean.
Step 4: Determine the critical value.
Since the alternative hypothesis is one-tailed (μ > 1.2), we need to find the critical value from the t-distribution with (n-1) degrees of freedom (df).
At a 0.05 significance level and 13 degrees of freedom (14-1), the critical value is approximately 1.771.
Step 5: Make a decision.
Compare the test statistic with the critical value.
If the test statistic is greater than the critical value, we reject the null hypothesis.
In this case, the test statistic (0/0.401) is less than the critical value (1.771).
Therefore, we fail to reject the null hypothesis.
Step 6: State the conclusion.
Based on the analysis, there is not enough evidence to conclude that the population mean volume of bottles is greater than 1.2 ml at the 0.05 significance level.