Understanding the t-Distribution and Hypothesis Testing

🔑 The distribution t is used when the sample size is small (n < 30).

📊 The t-table is used to determine the critical value based on the significance level and degrees of freedom.

🧪 The t-distribution is used to estimate the mean, test hypotheses, and determine the acceptance region.

📊 The transcript discusses the use of the t-distribution table in statistical analysis.

📈 The video explains how to calculate the t-value using the formula t = (x̄ - μ) / (s / √n).

📝 A specific example is given, where the average time for students to fill out a form is 50 minutes with a standard deviation.

⏱️ On average, it takes 42 minutes for 12 students to fill out the KRS form, with a standard deviation of 11.9 minutes.

💻 The hypothesis is that using a computer can speed up the KRS filling process.

📉 The significance level is set at 5%, indicating that if the filling time is faster than 50 minutes, the hypothesis is supported.

The video discusses the distribution of t with n less than 30.

The hypothesis is that the mean population is less than 50 minutes.

Using the provided formulas, the calculated value is -2.303.

🔍 The t-distribution is symmetric and has two tails.

✔️ The t-table is located on the left side of zero.

✅ Values in the rejection region of the t-table are greater than a certain value.

📚 When the value of t is less than 30, the critical region for h0 lies on the left side.

📈 If the calculated t-value falls within the critical region, h0 is rejected.

📉 If the calculated t-value is greater than the specified t-table value, h0 is accepted.

✅ The acceptance region signifies that h0 is accepted.

❌ The rejection region indicates that h0 is rejected.

🔍 In the given context, if h0 is accepted, it means that the conclusion is valid.

💡 If the calculated t-value falls within the critical region, it suggests that the conclusion is invalid.

⚠️ The t-value being less than 50 minutes implies a left-tailed distribution.

📊 The t-test value is negative 1.7, which falls in the rejection region of the null hypothesis.

✅ Therefore, the null hypothesis is rejected and the alternative hypothesis is accepted.

⏭️ The t-test value is smaller than the critical value, indicating that the new computer system is significantly faster than the old system.