The average weight of a jelly bean is known to be 1.10 grams. It’s known to have a standard deviation of 0.2 grams. 100 jelly beans have been weighted and found an average weight of 1.25 grams. Is there evidence that the known weight is true? Test it with a 0.05 level of significance.
The Correct Answer and Explanation is :
To determine if there is enough evidence to reject the null hypothesis (that the known average weight of a jelly bean is 1.10 grams), we can perform a hypothesis test. Since we know the population standard deviation (0.2 grams) and the sample size is large (100 jelly beans), we’ll use a Z-test for the population mean.
Hypothesis Setup:
- Null hypothesis (H₀): The population mean weight of a jelly bean is 1.10 grams, i.e., μ = 1.10.
- Alternative hypothesis (H₁): The population mean weight of a jelly bean is not 1.10 grams, i.e., μ ≠ 1.10.
Test Statistic Calculation:
The test statistic for a Z-test is given by the formula:
[
Z = \frac{\bar{X} – \mu}{\frac{\sigma}{\sqrt{n}}}
]
Where:
- (\bar{X}) = sample mean = 1.25 grams
- (\mu) = population mean (under H₀) = 1.10 grams
- (\sigma) = population standard deviation = 0.2 grams
- (n) = sample size = 100
Substituting the values:
[
Z = \frac{1.25 – 1.10}{\frac{0.2}{\sqrt{100}}} = \frac{0.15}{0.02} = 7.5
]
Decision Rule:
We are conducting a two-tailed test at a 0.05 level of significance, meaning the critical value for a Z-test is approximately ±1.96 (from Z-tables).
- If the absolute value of the test statistic (|Z|) is greater than 1.96, we reject the null hypothesis.
- If the absolute value of the test statistic (|Z|) is less than or equal to 1.96, we fail to reject the null hypothesis.
Conclusion:
The calculated Z-value is 7.5, which is much greater than 1.96. Therefore, we reject the null hypothesis.
Interpretation:
There is strong evidence at the 0.05 level of significance to suggest that the average weight of a jelly bean is not 1.10 grams. The sample mean of 1.25 grams significantly differs from the expected population mean of 1.10 grams. Thus, it appears that the known average weight of 1.10 grams is not accurate based on this sample.