Question: A bottle factory wants to test the hardness of their bottles. For a batch of 1,000 bottles, the average hardness should be greater than 11. They took a simple random sample of 15 bottles for destructive experiment and found their average hardness as 11.8 with a SD of 1.5. Can we say with a reasonable degree of certainty that the hardness of all 1,000 bottles is more than 11 (in other words, this is statistically significant) or can this be explained by chance?

Null Hypothesis: \[{{H}_{0}}:\mu =11\]

Alternative Hypothesis: \[{{H}_{A}}:\mu >11\]

Test statistic: t = 2.0656

P-value: 0.0289

Conclusion: We have enough evidence to support the claim that the hardness of all 1,000 bottles is more than 11, at the 0.05 significance level.

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