Find the critical value t α 2 t_{\frac{\alpha}{2}} 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 2, 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 2, and we also need to know our degrees of freedom. Now alpha over 2 is, of course, equal to 1 minus our confidence level. In this case, that's 80%, 0.8 divided by 2. So this gives me here 0.1. Then our degrees of freedom are n minus 1. So taking that 51 sample size and subtracting 1, 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 2 is then equal to 1.299. Let us know if you have any questions, and let's keep practicing.

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1. Intro to Stats and Collecting Data1h 14m
2. Describing Data with Tables and Graphs1h 55m
3. Describing Data Numerically2h 5m
4. Probability2h 16m
5. Binomial Distribution & Discrete Random Variables3h 6m
6. Normal Distribution and Continuous Random Variables2h 11m
7. Sampling Distributions & Confidence Intervals: Mean3h 23m
8. Sampling Distributions & Confidence Intervals: Proportion1h 12m
- Sampling Distribution of Sample Proportion
  29m
- Confidence Intervals for Population Proportion
  42m
9. Hypothesis Testing for One Sample3h 29m
10. Hypothesis Testing for Two Samples4h 50m
11. Correlation1h 6m
12. Regression1h 50m
13. Chi-Square Tests & Goodness of Fit1h 57m
14. ANOVA1h 57m

7. Sampling Distributions & Confidence Intervals: Mean

Confidence Intervals for Population Mean

Statistics

7. Sampling Distributions & Confidence Intervals: Mean

Confidence Intervals for Population Mean

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Multiple Choice

Find the critical value $t sub alpha over 2$ for an 80% confidence interval given a sample size of 51.

0.10

1.299

0.300

2.598

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Verified step by step guidance

Step 1: 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.

Step 2: Determine the degrees of freedom for the t-distribution. The degrees of freedom is calculated as the sample size minus one. For a sample size of 51, the degrees of freedom is 50.

Step 3: Identify the level of significance \alpha for the confidence interval. An 80% confidence interval implies that \alpha = 1 - 0.80 = 0.20.

Step 4: Calculate \frac{\alpha}{2} to find the critical value for a two-tailed test. \frac{\alpha}{2} = \frac{0.20}{2} = 0.10.

Step 5: Use a t-distribution table or calculator to find the critical value t_{\frac{\alpha}{2}} for \frac{\alpha}{2} = 0.10 and 50 degrees of freedom. This value is the critical t-value needed for the confidence interval.

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