A teacher gives her students the same test at the beginning and end of the week. She wants to know if their scores increase over time. The summary statistics that she has collected are below. $$ \bar{x}_{d}=4.2 $$ $$ s_{d}=4.3 $$ $$ n=35 $$ $$ H_{o}: \mu_{d}=0 \quad H_{a}: \mu_{d}0 $$ Given the summary statistics and hypotheses above, compute the $$ t $$ test statistic. Round your final answer to 3 decimal places.

Question

A teacher gives her students the same test at the beginning and end of the week. She wants to know if their scores increase over time. The summary statistics that she has collected are below. $$ \bar{x}_{d}=4.2 $$ $$ s_{d}=4.3 $$ $$ n=35 $$ $$ H_{o}: \mu_{d}=0 \quad H_{a}: \mu_{d}>0 $$ Given the summary statistics and hypotheses above, compute the $$ t $$ test statistic. Round your final answer to 3 decimal places.

UpStudy AI · Accepted Answer

To compute the $$ t $$ test statistic, we can use the formula:

$$ t = \frac{\bar{x}_{d} - \mu_{d}}{\frac{s_{d}}{\sqrt{n}}} $$

Given:
- $$ \bar{x}_{d} = 4.2 $$
- $$ s_{d} = 4.3 $$
- $$ n = 35 $$
- $$ H_{o}: \mu_{d} = 0 $$
- $$ H_{a}: \mu_{d} > 0 $$

Substitute the given values into the formula:

$$ t = \frac{4.2 - 0}{\frac{4.3}{\sqrt{35}}} $$

Now, we can calculate the $$ t $$ test statistic.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(4.2-0\right)}{\left(\frac{4.3}{\sqrt{35}}\right)}$$
- step1: Remove the parentheses:
  $$\frac{4.2-0}{\frac{4.3}{\sqrt{35}}}$$
- step2: Divide the terms:
  $$\frac{4.2-0}{\frac{43}{10\sqrt{35}}}$$
- step3: Remove 0:
  $$\frac{4.2}{\frac{43}{10\sqrt{35}}}$$
- step4: Convert the expressions:
  $$\frac{\frac{21}{5}}{\frac{43}{10\sqrt{35}}}$$
- step5: Multiply by the reciprocal:
  $$\frac{21}{5}\times \frac{10\sqrt{35}}{43}$$
- step6: Reduce the numbers:
  $$21\times \frac{2\sqrt{35}}{43}$$
- step7: Multiply:
  $$\frac{21\times 2\sqrt{35}}{43}$$
- step8: Multiply:
  $$\frac{42\sqrt{35}}{43}$$
The $$ t $$ test statistic is approximately 5.778.

Therefore, the $$ t $$ test statistic is approximately 5.778.
The $$ t $$ test statistic is approximately 5.778.

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