Estimation with Small Sample Size

Question

1 For a confidence level of 98% with a sample size of 24, find the critical t value.

2

If n=19, 

(x-bar)=49, and s=8, find the margin of error at a 90% confidence level

Give your answer to two decimal places.

3

Express the confidence interval 

 in interval form.

Express the answer in either decimal format or as percents.

Answer 1

Hi Marcella,

1) To find the critical t value, we need the degrees of freedom (df) which is n-1, so here it's 24-1=23. Then, depending on which t-table you use, we need the confidence (here it's .98) or the two tail area which is the same as 1-c, so .02. Then we look in the chart at the intersection for degrees of freedom 23 and two tail area of .02, which gives us the t critical value of 2.500.

2) The margin of error formula for a sample is E=(T_c•s)/(√n). We have to figure out our T_c, or T-critical, for this problem. So we figure out our df, which here is 19-1=18, and our two tail area, which here is 1-.9=.10, and see what number those intersect at, which gives us 1.734 as our T-critical. Now, we plug it into our formula: E= (1.734•8)/(√19)≈3.182. Our margin of error for this problem is 3.18 (rounded to two decimal places).

3) For interval form, we simply subtract the E (in this case 76) from our mean (638.5) and add it to our mean, and put the lower number and higher number in parentheses separated by a comma: (638.5-76,638.5+76)→(562.5, 714.5).

Hope this helps,

Ari