a. Does the sample mean differ significantly frome? b. Can it be said that the sample is above average? The owner of a factory that sells a particular bottled fruit juice claims that the average such bottles, calculates of each bottle, and then finds the mean capacity to be 248 ml . The standard deviation \( s \) is 15 ml . Is the claim true?

We wish to test whether the filling process is as claimed. Although the wording is a bit ambiguous, a common formulation of this type of problem is as follows. Suppose the manufacturer claims that the average volume is \[ \mu = 250\text{ ml}, \] but a sample of bottles yields a sample mean \[ \bar{x} = 248\text{ ml}, \] with a sample standard deviation \[ s = 15\text{ ml}. \] In what follows we assume that the sample size is \( n \) (for example, \( n = 36 \) is a common choice in such exercises). Under this assumption we address the two parts. --- **(a) Does the sample mean differ significantly from the claimed mean?** We set up the hypotheses for a two–tailed test as: \[ H_0: \mu = 250 \quad \text{versus} \quad H_1: \mu \neq 250. \] The test statistic is given by \[ t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}, \] where \(\mu_0 = 250\) ml. Assuming \( n = 36 \), we have \[ t = \frac{248 - 250}{15/\sqrt{36}} = \frac{-2}{15/6} = \frac{-2}{2.5} = -0.8. \] At a typical significance level (e.g., \(\alpha = 0.05\)) the critical values for a two–tailed \( t \)-test with \( n - 1 = 35 \) degrees of freedom are approximately \(\pm 2.03\). Since \[ -2.03 < -0.8 < 2.03, \] the test statistic does not fall into the rejection region. Thus, we do not reject \( H_0 \); there is no statistically significant difference between the sample mean and the claimed \(250\) ml. --- **(b) Can it be said that the sample is above average?** Now suppose we wish to test whether the filling is “above average,” i.e. that the true mean is higher than the claimed value. We then set up a one–tailed test: \[ H_0: \mu = 250 \quad \text{versus} \quad H_1: \mu > 250. \] The same test statistic remains \[ t = -0.8. \] For a one–tailed test at \(\alpha = 0.05\), the critical value is approximately \( t_{0.05,35} \approx 1.69 \). Since \[ -0.8 < 1.69, \] there is no evidence to support the claim that the actual average exceeds \(250\) ml. In fact, the sample mean is below \(250\) ml. --- **Summary of Answers:** - (a) The sample mean does not differ significantly from \(250\) ml. - (b) There is no evidence to conclude that the sample is above average; the sample mean of \(248\) ml is not greater than \(250\) ml. *Note:* If the problem statement intended a different claimed value or sample size, the same approach can be applied. The choice \( n = 36 \) was assumed here for demonstration purposes.