Math / Data & Statistics

Question```latex A health organization would like to determine if there is a difference in COVID mean recovery times between individuals who are 30 years old or younger and individuals who are older than 30 years old. The organization randomly selects 45 individuals who are 30 years old or younger who recovered from COVID and 50 individuals who are older than 30 years old who recovered from COVID-19. The table below summarizes the sample results for each group:
\begin{tabular}{|l|l|l|l|} \hline & Sample Size & Mean & Standard Deviation \\ \hline 30 years old or younger & 45 & 13 \text{ days} & 15 \text{ days} \\ \hline Older than 30 years Old & 50 & 3 \text{ days} & 2 \text{ and } 3 \text{ days} \\ \hline \end{tabular}
Test the claim that there is a difference in COVID mean recovery times between individuals who are 30 years old or younger and individuals who are older than 30 years old at a 5\% significance level.
D. State the null and alternative hypothesis.
E. State the P-value.
F. State your conclusion.

Studdy Solution

STEP 1

1. We are comparing the mean recovery times of two independent groups.
2. The sample sizes are large enough to use a t-test for independent samples.
3. The significance level is set at $\alpha = 0.05$ .
4. We assume that the samples are random and the populations are normally distributed or the sample sizes are large enough for the Central Limit Theorem to apply.

STEP 2

1. State the null and alternative hypotheses.
2. Calculate the test statistic.
3. Determine the P-value.
4. Make a decision based on the P-value and state the conclusion.

STEP 3

State the null and alternative hypotheses.
- Null Hypothesis ( $H_0$ ): There is no difference in mean recovery times between the two groups. Mathematically, $\mu_1 = \mu_2$ . - Alternative Hypothesis ( $H_a$ ): There is a difference in mean recovery times between the two groups. Mathematically, $\mu_1 \neq \mu_2$ .

STEP 4

Calculate the test statistic using the formula for the t-test for two independent samples:
$t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$
Where: - $\bar{x}_1 = 13$ , $s_1 = 15$ , $n_1 = 45$ - $\bar{x}_2 = 3$ , $s_2 = 2.3$ , $n_2 = 50$
$t = \frac{13 - 3}{\sqrt{\frac{15^2}{45} + \frac{2.3^2}{50}}}$
Calculate the values:
$t = \frac{10}{\sqrt{\frac{225}{45} + \frac{5.29}{50}}}$
$t = \frac{10}{\sqrt{5 + 0.1058}}$
$t = \frac{10}{\sqrt{5.1058}}$
$t \approx \frac{10}{2.26} \approx 4.42$

STEP 5

Determine the P-value using a t-distribution table or calculator for a two-tailed test with degrees of freedom approximated by:
$df = \frac{\left( \frac{s_1^2}{n_1} + \frac{s_2^2}{n_2} \right)^2}{\frac{\left( \frac{s_1^2}{n_1} \right)^2}{n_1 - 1} + \frac{\left( \frac{s_2^2}{n_2} \right)^2}{n_2 - 1}}$
Calculate the degrees of freedom:
$df \approx \frac{\left( \frac{225}{45} + \frac{5.29}{50} \right)^2}{\frac{\left( \frac{225}{45} \right)^2}{44} + \frac{\left( \frac{5.29}{50} \right)^2}{49}}$
$df \approx \frac{5.1058^2}{\frac{25}{44} + \frac{0.1058^2}{49}}$
$df \approx 69.5$
Using a t-table or calculator, find the P-value for $t = 4.42$ with $df \approx 69.5$ .

STEP 6

State your conclusion based on the P-value.
- If the P-value is less than $\alpha = 0.05$ , reject the null hypothesis. - If the P-value is greater than $\alpha = 0.05$ , fail to reject the null hypothesis.
Assuming the P-value is very small (as expected with a large t-value like 4.42), we reject the null hypothesis.
Conclusion: There is significant evidence to suggest that there is a difference in COVID mean recovery times between individuals who are 30 years old or younger and individuals who are older than 30 years old.

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