A study conducted stating does the temperature conditions affect the growth of height 

in children. The data was collected from two different locations A and location B. Check 

whether the external condition of temperature is affecting the growth or not at a level of 

significance of 5%.

A 98 102 96 104 100

B 100 105 98 102 104

To determine whether the external condition of temperature is affecting the growth of children's height at a 5% level of significance, we can use a hypothesis test. Since we are comparing the heights from two different locations (A and B), we can perform a **two-sample t-test** for the means.

### Steps to Perform a Two-Sample t-Test:

1. **State the Hypotheses**:
   - **Null Hypothesis (H₀)**: There is no significant difference between the mean heights of children from locations A and B (i.e., temperature does not affect growth).
     \[
     H₀: \mu_A = \mu_B
     \]
   - **Alternative Hypothesis (H₁)**: There is a significant difference between the mean heights of children from locations A and B (i.e., temperature affects growth).
     \[
     H₁: \mu_A \neq \mu_B
     \]

2. **Calculate the Means and Standard Deviations**:
   - Heights from location A: 98, 102, 96, 104, 100
   - Heights from location B: 100, 105, 98, 102, 104

   We will compute the means (\(\bar{x}_A\) and \(\bar{x}_B\)) and the standard deviations (\(s_A\) and \(s_B\)) for both samples.

3. **Determine the Degrees of Freedom** and **Compute the t-Statistic** using the formula:
   \[
   t = \frac{\bar{x}_A - \bar{x}_B}{\sqrt{\frac{s_A^2}{n_A} + \frac{s_B^2}{n_B}}}
   \]
   where:
   - \(n_A\) and \(n_B\) are the sample sizes for location A and B, respectively.
   - \(s_A^2\) and \(s_B^2\) are the variances (square of the standard deviations).

4. **Find the Critical Value** for a two-tailed test at the 5% level of significance.

5. **Make a Decision**:
   - If the computed t-value is greater than the critical value, reject the null hypothesis.
   - Otherwise, fail to reject the null hypothesis.

I'll now compute these values to perform the test.### Results of the Two-Sample t-Test:

1. **Means**:
   - Location A: Mean height = 100.0
   - Location B: Mean height = 101.8

2. **Standard Deviations**:
   - Location A: Standard deviation = 3.16
   - Location B: Standard deviation = 2.86

3. **t-Statistic**: \( t = -0.943 \)

4. **p-Value**: \( p = 0.373 \)

5. **Critical t-Value**: For a 5% significance level and degrees of freedom ≈ 7.92, the critical t-value is approximately \( t_{\text{critical}} = 2.31 \).

### Conclusion:
- Since the absolute value of the computed t-statistic (\(|t| = 0.943\)) is less than the critical value (2.31), and the p-value (0.373) is greater than 0.05, **we fail to reject the null hypothesis**.
- Therefore, there is **no statistically significant evidence** to suggest that the external temperature conditions are affecting the growth of children's height at the 5% significance level.

Would you like further details on any of these steps or concepts?

Here are 5 related questions to explore further:
1. What assumptions are made in a two-sample t-test, and are they satisfied here?
2. How would the results change if the sample sizes were larger?
3. What is the difference between a paired and an unpaired t-test?
4. How would a one-tailed test differ from this two-tailed test?
5. Can other factors besides temperature affect the height difference in children?

**Tip**: Always check the sample size and variance before choosing a t-test, as it influences the assumptions about equal or unequal variances.

A study conducted stating does the temperature conditions affect the growth of height in children. The data was collected from two different locations A and location B. Check whether the external condition of temperature is affecting the growth or not at a level of significance of 5%. A 98 102 96 104 100 B 100 105 98 102 104

This analysis determines whether temperature affects children's height growth by comparing data from two locations using a two-sample t-test at a 5% significance level. Learn key concepts in hypothesis testing, standard deviations, and p-values, suitable for students in Grades 10-12.

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