In an advertisement, a shoe store claims that its mean online delivery time is less than 30 hours. A random selection of 36 delivery times has a sample mean of 28.5 hours and a standard deviation of 3.5 hours.
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To determine whether the shoe store's claim is statistically supported, we need to perform a hypothesis test.
1. State the hypotheses:
- Null hypothesis (H₀): The mean online delivery time is 30 hours or more (μ ≥ 30).
- Alternative hypothesis (H₁): The mean online delivery time is less than 30 hours (μ < 30).
2. Set the significance level (α): This determines the threshold for accepting or rejecting the null hypothesis. Let's say α = 0.05 (5% level of significance).
3. Calculate the test statistic: In this case, we'll use the t-test since we don't know the population standard deviation.
The formula for the t-test statistic is:
t = (sample mean - hypothesized mean) / (sample standard deviation / sqrt(sample size))
t = (28.5 - 30) / (3.5 / sqrt(36))
4. Determine the degrees of freedom: The degrees of freedom is equal to the sample size minus 1 (df = n - 1). In this case, df = 36 - 1 = 35.
5. Find the critical value: Since the alternative hypothesis is μ < 30, we need to find the critical value from the left tail of the t-distribution with 35 degrees of freedom and α = 0.05.
Using a t-table or statistical software, the critical value for α = 0.05 and df = 35 is approximately -1.689.
6. Compare the test statistic with the critical value: If the test statistic is less than the critical value, we reject the null hypothesis; otherwise, we fail to reject it.
If t < -1.689, reject H₀. Otherwise, fail to reject H₀.
7. Calculate the p-value: The p-value is the probability of obtaining a test statistic as extreme (or more extreme) than the observed value, assuming the null hypothesis is true. We'll compare the p-value with our significance level (α) to determine statistical significance.
Using the t-distribution and the test statistic calculated earlier, we can find the p-value associated with that test statistic.
8. Make a conclusion: Based on the comparison of the test statistic and the critical value (or the p-value and the significance level), we can draw a conclusion.
- If the test statistic is less than the critical value or the p-value is less than α, we reject the null hypothesis. This would suggest that the mean online delivery time is statistically less than 30 hours, supporting the shoe store's claim.
- If the test statistic is greater than or equal to the critical value or the p-value is greater than or equal to α, we fail to reject the null hypothesis. This would indicate that there is not enough evidence to support the shoe store's claim.
Remember to perform these calculations using appropriate statistical software or a calculator for accuracy.