Answer :
Meter Sticks: Process standard deviation > 2 mm, confirmed at 1% level using chi-squared test.
Neptunian Weights: At 1% level, males outweigh females; t-test statistic > critical value.
How to test hypotheses?
Let's break down the two scenarios:
Meter Sticks:
Given data:
Sample mean = 1001.8 mm
Sample standard deviation (s) = 2.6 mm
Sample size (n) = 19
Population standard deviation claim (σ) = 2 mm (hypothesized value)
Hypotheses:
Null hypothesis (H0): σ = 2 mm
Alternative hypothesis (Ha): σ > 2 mm
We are testing if the population standard deviation is greater than 2 mm. Since the sample size is small (n < 30) and the population standard deviation is unknown, we should use the chi-squared distribution.
Calculate the chi-squared test statistic:
χ² = ((n - 1) * s²) / σ² = ((19 - 1) * 2.6²) / 2² ≈ 53.947
Using a chi-squared table or calculator, find the critical value for a 1% level of significance with 18 degrees of freedom (n - 1). The critical value is approximately 37.566.
Since the calculated test statistic (χ² ≈ 53.947) is greater than the critical value (37.566), we reject the null hypothesis.
Therefore, at a 1% level of significance, we can assert that the standard deviation of the process is over 2 mm.
Neptunian Weights:
Sample mean for males = 422 lbs.
Sample mean for females = 409 lbs.
Sample standard deviation (s) for males = 30 lbs.
Sample standard deviation (s) for females = 28 lbs.
Sample size for males (n1) = 92
Sample size for females (n2) = 103
Hypotheses:
Null hypothesis (H0): μ1 - μ2 ≤ 0 (No significant difference)
Alternative hypothesis (Ha): μ1 - μ2 > 0 (Males weigh more than females)
We are testing if male Neptunians weigh more than female Neptunians. Since we have two independent samples and population standard deviations are unknown, we should use the t-distribution.
Calculate the test statistic:
t = (mean1 - mean2) / √((s1²/n1) + (s2²/n2)) = (422 - 409) / √((30²/92) + (28²/103)) ≈ 2.961
Using a t-distribution table or calculator with appropriate degrees of freedom, find the critical t-value for a 1% level of significance (1-tailed). The critical value for 1% significance with degrees of freedom around 190 is approximately 2.613.
Since the calculated test statistic (t ≈ 2.961) is greater than the critical value (2.613), we reject the null hypothesis.
Therefore, at a 1% level of significance, we can assert that, in general, male Neptunians weigh more than female Neptunians.
Learn more about hypotheses on:
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