Question

A) Construct a 95% confidence interval for the average test score for Delhi students. (1 Mark)

Answer 1

The solution to all the answers are given below.

The complete question includes

Grades on a standardized test are known to have a mean of 1,000 for students in the Delhi. 453

randomly selected Delhi students take the test, yielding sample mean of 1,013 and sample standard

deviation (s) of 108.

What is Confidence Interval ?

It is given by

Confidence Interval for 95% confidence Interval is given by

$\rm Z = X \pm 1.96 \dfrac{\sigma}{\sqrt{n}}$

(a) Construct a 95% confidence interval for the mean test score for Delhi students.

The confidence interval is given by

$\rm 1,013 \pm 1.96 \dfrac{\sigma}{\sqrt{n}}$

$\rm 1,013 \pm 1.96 \dfrac{108}{\sqrt{453}}$

1013 ± 5.07

so the interval is  [1003.06,1022.94]

(b)Yes, since the null of no difference is rejected at the 5% significance level (interval excludes Delhi sample mean of 1,013)

(c) Another 503 Delhi students are randomly selected to take a 3-hour prep course and then give the test. Their average score is 1,019 with a standard deviation of 95.

The standard deviation now is

$\rm \sqrt{\dfrac{95^2}{453} + \dfrac{108^2}{503}}$

= 6.61

The interval is given by

$(1,019-1,013) \pm 1.96 \dfrac{\sigma}{\sqrt{n}}$

= [-7,+19]

(d) No, the interval includes 0, the null difference between the two populations

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