Answer :
To test the claim that the two samples are from populations with the same mean, we need to follow these steps:
1. State the hypotheses:
- The null hypothesis ([tex]\(H_0\)[/tex]): The means of the two populations are equal.
- The alternative hypothesis ([tex]\(H_1\)[/tex]): The means of the two populations are not equal.
These hypotheses can be written as:
- [tex]\( H_0: \mu_1 = \mu_2 \)[/tex]
- [tex]\( H_1: \mu_1 \neq \mu_2 \)[/tex]
Therefore, the correct choice is:
- B. [tex]\( H_0: \mu_1 = \mu_2 \)[/tex] and [tex]\( H_1: \mu_1 \neq \mu_2 \)[/tex]
2. Calculate the test statistic:
- The test statistic [tex]\( t_c \)[/tex] measures how far the sample means are from each other, taking into account the sample sizes and standard deviations.
Given values:
- Sample sizes: [tex]\( n_1 = 31 \)[/tex] and [tex]\( n_2 = 37 \)[/tex]
- Sample means: [tex]\( x_1 = 2.31 \)[/tex] and [tex]\( x_2 = 2.69 \)[/tex]
- Sample standard deviations: [tex]\( s_1 = 0.84 \)[/tex] and [tex]\( s_2 = 0.58 \)[/tex]
The formula for the test statistic [tex]\( t_c \)[/tex] for two independent samples is:
[tex]\[ t_c = \frac{(x_1 - x_2)}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} \][/tex]
Plugging in the values:
[tex]\[ t_c = \frac{(2.31 - 2.69)}{\sqrt{\frac{0.84^2}{31} + \frac{0.58^2}{37}}} \][/tex]
Using these numbers, the calculated test statistic [tex]\( t_c \)[/tex] is approximately:
- [tex]\( t_c = -2.13 \)[/tex]
3. Determine the degrees of freedom:
- Degrees of freedom (df) can be complex to calculate as they involve the variances and sample sizes of both groups.
- Using the approximation formula for degrees of freedom for unequal variances:
[tex]\[ df = \frac{(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2})^2} {\frac{(\frac{s_1^2}{n_1})^2}{n_1 - 1} + \frac{(\frac{s_2^2}{n_2})^2}{n_2 - 1}} \][/tex]
Using the given values, the degrees of freedom (df) is approximately:
- [tex]\( df \approx 36.79 \)[/tex]
4. Determine the critical value:
- For a two-tailed test at a significance level of [tex]\( \alpha = 0.01 \)[/tex], we look up the t-distribution table for the critical value corresponding to [tex]\( \alpha/2 = 0.005 \)[/tex] and 36.79 degrees of freedom.
- The critical value for this t-distribution is approximately:
- Critical value [tex]\( \approx 2.68 \)[/tex]
5. Calculate the p-value:
- The p-value indicates the probability of obtaining a test statistic at least as extreme as the one calculated assuming the null hypothesis is true. For [tex]\( t_c = -2.13 \)[/tex], and df approximated, the p-value is:
[tex]\[ p\_value \approx 0.0381 \][/tex]
In summary:
- The hypotheses are:
- [tex]\( H_0: \mu_1 = \mu_2 \)[/tex]
- [tex]\( H_1: \mu_1 \neq \mu_2 \)[/tex]
- The test statistic [tex]\( t_c = -2.13 \)[/tex]
- The critical value for a two-tailed test at [tex]\( \alpha = 0.01 \)[/tex] is [tex]\( \pm 2.68 \)[/tex]
- The p-value is approximately [tex]\( 0.0381 \)[/tex]
Since the p-value (0.0381) is greater than the significance level [tex]\( \alpha = 0.01 \)[/tex], we do not reject the null hypothesis. Thus, there is insufficient evidence to conclude that the population means are different.
1. State the hypotheses:
- The null hypothesis ([tex]\(H_0\)[/tex]): The means of the two populations are equal.
- The alternative hypothesis ([tex]\(H_1\)[/tex]): The means of the two populations are not equal.
These hypotheses can be written as:
- [tex]\( H_0: \mu_1 = \mu_2 \)[/tex]
- [tex]\( H_1: \mu_1 \neq \mu_2 \)[/tex]
Therefore, the correct choice is:
- B. [tex]\( H_0: \mu_1 = \mu_2 \)[/tex] and [tex]\( H_1: \mu_1 \neq \mu_2 \)[/tex]
2. Calculate the test statistic:
- The test statistic [tex]\( t_c \)[/tex] measures how far the sample means are from each other, taking into account the sample sizes and standard deviations.
Given values:
- Sample sizes: [tex]\( n_1 = 31 \)[/tex] and [tex]\( n_2 = 37 \)[/tex]
- Sample means: [tex]\( x_1 = 2.31 \)[/tex] and [tex]\( x_2 = 2.69 \)[/tex]
- Sample standard deviations: [tex]\( s_1 = 0.84 \)[/tex] and [tex]\( s_2 = 0.58 \)[/tex]
The formula for the test statistic [tex]\( t_c \)[/tex] for two independent samples is:
[tex]\[ t_c = \frac{(x_1 - x_2)}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} \][/tex]
Plugging in the values:
[tex]\[ t_c = \frac{(2.31 - 2.69)}{\sqrt{\frac{0.84^2}{31} + \frac{0.58^2}{37}}} \][/tex]
Using these numbers, the calculated test statistic [tex]\( t_c \)[/tex] is approximately:
- [tex]\( t_c = -2.13 \)[/tex]
3. Determine the degrees of freedom:
- Degrees of freedom (df) can be complex to calculate as they involve the variances and sample sizes of both groups.
- Using the approximation formula for degrees of freedom for unequal variances:
[tex]\[ df = \frac{(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2})^2} {\frac{(\frac{s_1^2}{n_1})^2}{n_1 - 1} + \frac{(\frac{s_2^2}{n_2})^2}{n_2 - 1}} \][/tex]
Using the given values, the degrees of freedom (df) is approximately:
- [tex]\( df \approx 36.79 \)[/tex]
4. Determine the critical value:
- For a two-tailed test at a significance level of [tex]\( \alpha = 0.01 \)[/tex], we look up the t-distribution table for the critical value corresponding to [tex]\( \alpha/2 = 0.005 \)[/tex] and 36.79 degrees of freedom.
- The critical value for this t-distribution is approximately:
- Critical value [tex]\( \approx 2.68 \)[/tex]
5. Calculate the p-value:
- The p-value indicates the probability of obtaining a test statistic at least as extreme as the one calculated assuming the null hypothesis is true. For [tex]\( t_c = -2.13 \)[/tex], and df approximated, the p-value is:
[tex]\[ p\_value \approx 0.0381 \][/tex]
In summary:
- The hypotheses are:
- [tex]\( H_0: \mu_1 = \mu_2 \)[/tex]
- [tex]\( H_1: \mu_1 \neq \mu_2 \)[/tex]
- The test statistic [tex]\( t_c = -2.13 \)[/tex]
- The critical value for a two-tailed test at [tex]\( \alpha = 0.01 \)[/tex] is [tex]\( \pm 2.68 \)[/tex]
- The p-value is approximately [tex]\( 0.0381 \)[/tex]
Since the p-value (0.0381) is greater than the significance level [tex]\( \alpha = 0.01 \)[/tex], we do not reject the null hypothesis. Thus, there is insufficient evidence to conclude that the population means are different.
