Find the critical value t α 2 t_{\frac{\alpha}{2}} t 2 α for an 80% confidence interval given a sample size of 51.

Here we're asked to find the critical t value, that's t alpha over two, or an 80% confidence interval given a sample size, that's n, of 51. Now remember that in order to find our critical t value, we need to know alpha over two, and we also need to know our degrees of freedom. Now alpha over two is of course equal to one minus our confidence level. In this case, that's 80%, zero point eight divided by two. So this gives me here 0.1. Then our degrees of freedom are n minus one. So taking that 51 sample size and subtracting one, this results in 50 degrees of freedom. Now from this point, you can either use a calculator or look this up in a t table. Either way, you should find that t alpha over two is then equal to 1.299. Let us know if you have any questions, and let's keep practicing.

7. Sampling Distributions & Confidence Intervals: Mean

Confidence Intervals for Population Mean

Statistics for Business

7. Sampling Distributions & Confidence Intervals: Mean

Confidence Intervals for Population Mean

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Multiple Choice

Find the critical value $t_{\frac{\alpha}{2}}$ for an 80% confidence interval given a sample size of 51.

0.10

1.299

0.300

2.598

Verified step by step guidance

Understand that the critical value t_{\frac{\alpha}{2}} is used in constructing confidence intervals for the mean when the population standard deviation is unknown and the sample size is small.

Identify that the confidence level is 80%, which means the significance level \alpha is 0.20 (since \alpha = 1 - confidence level).

Calculate \frac{\alpha}{2} to find the tail probability for the t-distribution. For an 80% confidence interval, \frac{\alpha}{2} = 0.10.

Determine the degrees of freedom for the t-distribution, which is the sample size minus one. For a sample size of 51, the degrees of freedom is 50.

Use a t-distribution table or a statistical software to find the critical value t_{\frac{\alpha}{2}} for \frac{\alpha}{2} = 0.10 and 50 degrees of freedom.

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Find the critical value tα2t_{\frac{\alpha}{2}}t2αfor an 80% confidence interval given a sample size of 51.

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Find the critical value $t_{\frac{\alpha}{2}}$ for an 80% confidence interval given a sample size of 51.