If You Don't Have Experience Using Minitab Software Please D

If You Dont Have Experience Using Minitab Software Please Do Not

*If you don't have experience using Minitab software, please do not attempt the following 2 problems: Solve Examples 1 and 2 using Minitab and interpret the output in each example to conclude whether the null hypothesis must be rejected. Please provide more than just the answer. Must be able to see work. Example 1 Problem 1 Jeffrey, as an eight-year old, established a mean time of 16.43 seconds for swimming the 25-yard freestyle, with a standard deviation of 0.8 seconds. His dad, Frank, thought that Jeffrey could swim the 25-yard freestyle faster by using goggles. Frank bought Jeffrey a new pair of expensive goggles and timed Jeffrey for 15 25-yard freestyle swims. For the 15 swims, Jeffrey's mean time was 16 seconds. Frank thought that the goggles helped Jeffrey to swim faster than the 16.43 seconds. Conduct a hypothesis test using a preset α = 0.05. Assume that the swim times for the 25-yard freestyle are normal. Example 2 Problem 1 A college football coach thought that his players could bench press a mean weight of 275 pounds. It is known that the standard deviation is 55 pounds. Three of his players thought that the mean weight was more than that amount. They asked 30 of their teammates for their estimated maximum lift on the bench press exercise. The data ranged from 205 pounds to 385 pounds. The actual different weights were (frequencies are in parentheses): 205(3), 215(3), 225(1), 241(2), 252(2), 265(2), 275(2), 313(2), 316(5), 338(2), 341(1), 345(2), 368(2), 385(1). Conduct a hypothesis test using a 2.5% level of significance to determine if the bench press mean is more than 275 pounds.

Paper For Above instruction

Hypothesis testing is a fundamental procedure in statistics, used to make inferences or draw conclusions about a population parameter based on sample data. It involves formulating a null hypothesis (H₀), which represents the default assumption to be tested, and an alternative hypothesis (H₁), which reflects the research question or the effect we suspect might exist. The decision to reject or fail to reject the null hypothesis hinges on the analysis of sample data, considering the significance level (α), which defines the threshold for acceptable evidence against H₀. In this paper, two specific examples are analyzed using hypothesis testing principles to illustrate the process and interpretation of results when using software like Minitab.

Example 1: Testing Jeffrey’s Swimming Speed

Jeffrey's swimming times are analyzed to determine whether the goggles have a statistically significant effect on his performance. Historically, Jeffrey's mean swim time was 16.43 seconds with a standard deviation of 0.8 seconds. The sample data from 15 swims with the goggles indicate a mean of 16 seconds. The research question is whether the use of goggles reduces Jeffrey's mean swim time below 16.43 seconds. The hypotheses are set as:

  • H₀: μ = 16.43 seconds (no improvement)
  • H₁: μ

Using a significance level of α = 0.05, the t-test for the mean is appropriate since the population standard deviation is unknown and the sample size is small. The test statistic is calculated as:

 t = (x̄ - μ₀) / (s / √n)

Where:

  • x̄ = 16 seconds (sample mean)
  • μ₀ = 16.43 seconds (historical mean)
  • s = 0.8 seconds (standard deviation)
  • n = 15 (sample size)

Substituting the values:

 t = (16 - 16.43) / (0.8 / √15) ≈ -2.563

Consulting Minitab or t-distribution tables with df = 14 (n-1) provides the critical value for a one-tailed test at α = 0.05, which is approximately -1.761. Since the calculated t-value (-2.563) is less than -1.761, the evidence suggests rejecting the null hypothesis. Therefore, there is statistically significant evidence that Jeffrey's swim time improved with goggles, reducing his mean time below 16.43 seconds.

Example 2: Testing the Mean Bench Press Weight

The second example involves determining whether the mean maximum lift of college football players exceeds 275 pounds. Known parameters include a standard deviation of 55 pounds. The sample data from 30 players cover a frequency distribution of weights ranging from 205 to 385 pounds, with various counts. The null hypothesis states that the mean is 275 pounds:

  • H₀: μ = 275 pounds
  • H₁: μ > 275 pounds

The significance level for this test is α = 0.025, or 2.5%. The sample mean is calculated based on the frequency distribution:

Sum of all weighted weights:



5 × 316 + 2 × 338 + 2 × 341 + 2 × 345 + 2 × 368 + other includes 205,215,...

Calculating the mean involves multiplying each weight by its frequency, summing, and dividing by the total number of observations (30). Doing this yields:



[(3×205) + (3×215) + (1×225) + (2×241) + (2×252) + (2×265) + (2×275) + (2×313) + (5×316) + (2×338) + (1×341) + (2×345) + (2×368) + (1×385)] / 30 ≈ 293.57 pounds

The test statistic for the mean is given by:

 z = (x̄ - μ₀) / (σ / √n)

where:

  • x̄ ≈ 293.57 pounds (sample mean)
  • μ₀ = 275 pounds (hypothesized mean)
  • σ = 55 pounds (known standard deviation)
  • n = 30

Calculating z:

 z = (293.57 - 275) / (55 / √30) ≈ 3.74

For a one-tailed test at α = 0.025, the critical z-value is approximately 1.96. Since 3.74 > 1.96, we reject the null hypothesis, concluding there is significant evidence that the mean bench press weight exceeds 275 pounds.

Conclusion

Both examples demonstrate the application of hypothesis testing in real-world scenarios: assessing a performance improvement in swimming and evaluating the average strength of athletes. The key steps include formulating hypotheses, calculating the appropriate test statistic, and comparing it to critical values based on the significance level. Using Minitab or similar statistical software simplifies calculations and provides detailed output, including confidence intervals and p-values, which aid in making informed decisions. These practice examples emphasize the importance of understanding the underlying principles of hypothesis testing beyond software use, highlighting the need for proper data analysis and interpretation in making valid statistical inferences.

References

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