The National Academy of Science reported that of research in mathematics is published by US authors. The mathematics chairperson of a
prestigious university wishes to test the claim that this percentage is no longer . He has no indication of whether the percentage has increased
or decreased since that time. He surveys a simple random sample of 280 recent articles published by reputable mathematics research journals and
finds that 108 of these articles have US authors. Does this evidence support the mathematics chairperson’s claim that the percentage is no longer
? Use a 0.01 level of significance.

Ask by Deleon Romero. in the United States

Nov 16,2024

Question

The National Academy of Science reported that of research in mathematics is published by US authors. The mathematics chairperson of a
prestigious university wishes to test the claim that this percentage is no longer . He has no indication of whether the percentage has increased
or decreased since that time. He surveys a simple random sample of 280 recent articles published by reputable mathematics research journals and
finds that 108 of these articles have US authors. Does this evidence support the mathematics chairperson’s claim that the percentage is no longer
? Use a 0.01 level of significance.

Ask by Deleon Romero. in the United States

Nov 16,2024

UpStudy AI · Accepted Answer

To test the claim that the percentage of research in mathematics published by US authors is no longer $$ 33\% $$, we can perform a hypothesis test for a proportion.

### Step 1: Define the Hypotheses

- Null Hypothesis ($$ H_0 $$): $$ p = 0.33 $$ (The proportion of articles by US authors is $$ 33\% $$)
- Alternative Hypothesis ($$ H_a $$): $$ p \neq 0.33 $$ (The proportion of articles by US authors is not $$ 33\% $$)

### Step 2: Collect the Sample Data

- Sample size ($$ n $$) = 280
- Number of articles with US authors ($$ X $$) = 108
- Sample proportion ($$ \hat{p} $$) = $$ \frac{X}{n} = \frac{108}{280} \approx 0.3857 $$

### Step 3: Calculate the Test Statistic

We will use the formula for the test statistic for a proportion:

$$
z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}
$$

where:
- $$ p_0 = 0.33 $$ (the hypothesized population proportion)
- $$ \hat{p} = 0.3857 $$ (the sample proportion)
- $$ n = 280 $$ (the sample size)

First, we calculate the standard error:

$$
SE = \sqrt{\frac{p_0(1 - p_0)}{n}} = \sqrt{\frac{0.33(1 - 0.33)}{280}} = \sqrt{\frac{0.33 \times 0.67}{280}} \approx \sqrt{\frac{0.2211}{280}} \approx \sqrt{0.0007904} \approx 0.0281
$$

Now, we can calculate the $$ z $$-score:

$$
z = \frac{0.3857 - 0.33}{0.0281} \approx \frac{0.0557}{0.0281} \approx 1.979
$$

### Step 4: Determine the Critical Value

Since this is a two-tailed test at the $$ \alpha = 0.01 $$ significance level, we need to find the critical values for $$ z $$.

Using a standard normal distribution table, the critical values for a two-tailed test at $$ \alpha = 0.01 $$ are approximately $$ z = \pm 2.576 $$.

### Step 5: Make a Decision

Now we compare the calculated $$ z $$-score with the critical values:

- If $$ |z| > 2.576 $$, we reject the null hypothesis.
- If $$ |z| \leq 2.576 $$, we fail to reject the null hypothesis.

In our case:

$$
|z| \approx 1.979 \leq 2.576
$$

### Conclusion

Since $$ |z| $$ is less than $$ 2.576 $$, we fail to reject the null hypothesis. There is not enough evidence at the $$ 0.01 $$ level of significance to support the mathematics chairperson's claim that the percentage of research in mathematics published by US authors is no longer $$ 33\% $$.
The test results fail to reject the null hypothesis, indicating insufficient evidence to support the claim that the percentage of research in mathematics published by US authors is no longer 33%.

Upstudy AI Solution

Answer

Solution

Bonus Knowledge

Conclusion:

Related Questions

Latest Statistics Questions