Additional Information And Full Hypothesis Test Examples

95 Additional Information And Full Hypothesis Test Examplesfor Each O

A particular brand of tires claims that its deluxe tire averages at least 50,000 miles before it needs to be replaced. From past studies of this tire, the standard deviation is known to be 8,000. A survey of owners of that tire design is conducted. From the 28 tires surveyed, the mean lifespan was 46,500 miles with a standard deviation of 9,800 miles. Using alpha = 0.05, is the data highly inconsistent with the claim?

From generation to generation, the mean age when smokers first start to smoke varies. However, the standard deviation of that age remains constant of around 2.1 years. A survey of 40 smokers of this generation was done to see if the mean starting age is at least 19. The sample mean was 18.1 with a sample standard deviation of 1.3. Do the data support the claim at the 5% level?

Paper For Above instruction

Hypothesis testing is a fundamental statistical method used to determine whether there is enough evidence to support a specific claim about a population parameter. In the context of the provided examples—testing a tire's mean lifespan and a smoker's mean starting age—hypothesis testing enables researchers to make informed decisions based on sample data and predefined significance levels.

### Example 1: Tire Lifespan

The claim of the tire company is that the average lifespan of their deluxe tires is at least 50,000 miles. To evaluate this claim, a hypothesis test is conducted with the following hypotheses:

  • Null hypothesis (H₀): μ ≥ 50,000 miles (the claim is true)
  • Alternative hypothesis (H₁): μ

Given that the population standard deviation (σ) is known (8,000 miles), and the sample size is 28, with a sample mean (x̄) of 46,500 miles, the test statistic is calculated using the z-test formula:

z = (x̄ - μ₀) / (σ / √n)

Substituting the values:

z = (46,500 - 50,000) / (8,000 / √28) ≈ -3,500 / (8,000 / 5.291) ≈ -3,500 / 1,512.86 ≈ -2.31

At a significance level of α = 0.05, the critical z-value for a one-tailed test is approximately -1.645. Since -2.31

### Example 2: Age at Smoking Initiation

The hypothesis here concerns whether the mean age at which smokers start is at least 19 years, given the population standard deviation remains constant at approximately 2.1 years. The hypotheses are:

  • Null hypothesis (H₀): μ ≥ 19 years
  • Alternative hypothesis (H₁): μ

The sample data include a sample size (n) of 40, a sample mean (x̄) of 18.1 years, and a sample standard deviation (s) of 1.3 years. Assuming the standard deviation is close to the population value, the z-test statistic is:

z = (x̄ - μ₀) / (σ / √n) = (18.1 - 19) / (2.1 / √40) ≈ -0.9 / (2.1 / 6.3246) ≈ -0.9 / 0.3324 ≈ -2.70

With α = 0.05, the critical z-value for a one-tailed test is -1.645. Since -2.70

### Conclusion

In hypotheses tests, the decision to reject or fail to reject the null hypothesis depends on the comparison of the test statistic with the critical value. If the test statistic falls within the critical region, the null hypothesis is rejected, implying the sample data support the alternative hypothesis. Conversely, if it does not, there is insufficient evidence to challenge the null hypothesis. In these examples, both tests led to rejecting the null hypotheses at the 5% significance level, suggesting the true means are below the claimed thresholds.

References

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  • Statistics Canada. (2018). Canadian Community Health Survey: Smoking. Government of Canada.
  • Walpole, R. E., Myers, R. H., Myers, S. L., & Ye, K. (2012). Probability & statistics for engineering and the sciences (9th ed.). Pearson.

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