Questions: For the standard deviation unknown case, what is the t value for a confidence of 90% when the sample size is 13?

Transcript text: For the standard deviation unknown case, what is the $t$ value for a confidence of $90 \%$ when the sample size is 13 ?

Solution

Solution Steps

To find the $t$ value for a confidence level of 90% with a sample size of 13, we need to use the t-distribution table or a statistical function in Python. The degrees of freedom (df) will be the sample size minus one.

Step 1: Determine Degrees of Freedom

For a sample size $ n = 13 $, the degrees of freedom $ df $ is calculated as: \[ df = n - 1 = 13 - 1 = 12 \]

Step 2: Find the Critical $ t $ Value

To find the critical $ t $ value for a confidence level of $ 90\% $, we use the formula: \[ t = t_{(1 - \alpha/2, df)} \] where $ \alpha = 1 - 0.90 = 0.10 $. Thus, we need to find: \[ t_{(0.05, 12)} \] The calculated value is: \[ t \approx 1.7823 \]

Final Answer

The $ t $ value for a confidence of $ 90\% $ when the sample size is $ 13 $ is approximately $ 1.7823 $. Therefore, the answer is: \[ \boxed{1.7823} \]

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