5. An Aspirin manufacturer fills bottles by weight rather than by count. Since each bottle should contain 100 tablets, the average weight per tablet should be 5 grains. Each of 100 tablets taken from a large lot is weighed, resulting in a sample average weight per tablet of 4.87 grains and a sample standard deviation of 0.35 grain. Does this information provide strong evidence for concluding that the company is not filling its bottles as advertised? Test H0: μ = 5 against HA: μ < 5 at the 0.01 level of significance
To determine if there is strong evidence for concluding that the company is not filling its bottles as advertised, we can conduct a hypothesis test using the given information.
H0: μ = 5 (null hypothesis: the average weight per tablet is 5 grains)
HA: μ < 5 (alternative hypothesis: the average weight per tablet is less than 5 grains)
We are given the sample average weight per tablet (x̄) = 4.87 grains and the sample standard deviation (s) = 0.35 grain. The sample size is 100 tablets.
To conduct the hypothesis test, we first need to calculate the test statistic. In this case, we will use a t-test since the population standard deviation is unknown.
The formula for the t-test statistic is:
t = (x̄ - μ) / (s / √n)
Where:
x̄ is the sample mean
μ is the population mean (hypothesized value)
s is the sample standard deviation
n is the sample size
Plugging in the given values, we have:
t = (4.87 - 5) / (0.35 / √100)
= (-0.13) / (0.35 / 10)
= (-0.13) / 0.035
= -3.71
Next, we need to find the critical value for the test. Since we are testing at the 0.01 level of significance and the alternative hypothesis is one-tailed (μ < 5), we need to find the critical t-value at the 0.01 level of significance with 99 degrees of freedom (n-1).
Using a statistical table or a calculator, we find the critical t-value to be approximately -2.626.
Comparing the test statistic (-3.71) to the critical value (-2.626), we find that the test statistic is more extreme (further in the left tail) than the critical value. This means we reject the null hypothesis.
Therefore, the given information does provide strong evidence for concluding that the company is not filling its bottles as advertised.
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