Question

Reading scores in Atlanta The Trial Urban District Assessment (TUDA) is a

government-sponsored study of student achievement in large urban school districts. TUDA gives a reading test scored from $0$to $500$. A score of $243$is a “basic” reading level and a score of $281$is “proficient.” Scores for a random sample of $1470$eighth-graders in Atlanta had a mean of $240$with standard deviation of $42.17.$

a. Construct and interpret a $99\%$confidence interval for the mean reading test score of all Atlanta eighth-graders.

b. Based on your interval from part (a), is there convincing evidence that the mean reading test score for all Atlanta eighth-graders is less than the basic level? Explain your answer.

Short answer

a. Confidence Interval is $(237.163,242.837)$

b. Yes, there is convincing evidence that the mean reading test score for all Atlanta eighth-graders is less than the basic level

Step-by-step solution

Given Information

It is given that $n=1470$

$\left(\overline{x}\right)=240$

$\left(s\right)=42.17$

Calculating Confidence Interval

It can be calculated as $CI=\overline{x}\pm t_{\alpha /2,df}\times \frac{s}{\sqrt{n}}$

As population standard deviation is not known, we use $t$confidence interval.

Output of Excel is:

Hence, confidence interval is $(237.163,242.837)$

So, there is $99\%$confidence that mean reading score is between$237.163\text{and}242.837.$

To check if mean reading score is less than basic level score or not (243)

Confidence Interval is $(237.163,242.837)$. Mean reading score is $243$. It lie between the interval $237.163\mathrm{and}242.837$.

Hence, we can say that mean reading score is less than basic level score.