A shoe maker company produces a specific model of shoes having 15 months average lifetime. One of the employees in their R & D division claims to have developed a product that lasts longer. This latest product was worn by 30 people and lasted on average for 17 months. The variability of the original shoe is estimated based on the standard deviation of the new group which is 5.5 months. Is the designer's claim of a better shoe supported by the findings of the trial? Make your decision using two tailed testing using a level of significance of p < .05.
Good
To determine whether the designer's claim of a better shoe is supported by the findings of the trial, we can use hypothesis testing. Let's define our null and alternative hypotheses:
Null Hypothesis (H0): The average lifetime of the designer's shoes is the same as the original shoes (μ = 15).
Alternative Hypothesis (H1): The average lifetime of the designer's shoes is longer than the original shoes (μ > 15).
We will use a two-tailed test because we want to determine if the new shoes have either a significantly shorter or significantly longer lifetime compared to the original shoes.
Next, let's calculate the test statistic (t-value) and compare it to the critical value. The formula for the t-value is:
t = (sample mean - population mean) / (sample standard deviation / √n)
Given information:
Original shoes average lifetime: μ = 15
Designer's shoes average lifetime: sample mean = 17
Sample standard deviation: σ = 5.5
Number of people: n = 30
Calculating the t-value:
t = (17 - 15) / (5.5 / √30)
t = 2 / (5.5 / √30)
Now, we need to find the critical value. Since this is a two-tailed test and the level of significance is p < 0.05, the critical values will be ± tα/2, df, where α = 0.05 and df = n - 1.
df = 30 - 1 = 29
tα/2,29 = ± 2.045 (from the t-distribution table)
If the calculated t-value is greater than 2.045 or less than -2.045, we will reject the null hypothesis in favor of the alternative hypothesis.
Now we can calculate the t-value and compare it to the critical value:
t = 2 / (5.5 / √30)
If the calculated t-value exceeds the critical value of 2.045 or if it is less than -2.045, we reject the null hypothesis in favor of the alternative hypothesis. Otherwise, we fail to reject the null hypothesis.
Note: If we reject the null hypothesis, it means that the designer's claim of a longer-lasting shoe is supported by the findings of the trial. If we fail to reject the null hypothesis, it means that there is not enough evidence to support the claim.