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To determine whether Type I bulbs are superior to Type II bulbs regarding their length of life, we will perform an independent samples t-test. This test will help us compare the means of the two independent samples to see if there is a statistically significant difference between them.
Here are the steps for this hypothesis test:
1. State the Hypotheses:
- Null Hypothesis [tex]\(H_0\)[/tex]: The mean length of life of Type I bulbs is equal to the mean length of life of Type II bulbs. [tex]\(\mu_1 = \mu_2\)[/tex]
- Alternative Hypothesis [tex]\(H_a\)[/tex]: The mean length of life of Type I bulbs is greater than the mean length of life of Type II bulbs. [tex]\(\mu_1 > \mu_2\)[/tex]
2. Significance Level:
- The significance level [tex]\(\alpha\)[/tex] is 0.05.
3. Calculate the Standard Error (SE):
The standard error of the difference in means is calculated using the formula:
[tex]\[ SE = \sqrt{\left(\frac{sd_1^2}{n_1}\right) + \left(\frac{sd_2^2}{n_2}\right)} \][/tex]
Using the given data:
- [tex]\(sd_1 = 36\)[/tex]
- [tex]\(n_1 = 50\)[/tex]
- [tex]\(sd_2 = 40\)[/tex]
- [tex]\(n_2 = 50\)[/tex]
[tex]\[ SE = \sqrt{\left(\frac{36^2}{50}\right) + \left(\frac{40^2}{50}\right)} = 7.610519036176179 \][/tex]
4. Calculate the Test Statistic:
The test statistic (t) is calculated using the formula:
[tex]\[ t = \frac{(\bar{x}_1 - \bar{x}_2)}{SE} \][/tex]
Where,
- [tex]\(\bar{x}_1 = 1234\)[/tex]
- [tex]\(\bar{x}_2 = 1215\)[/tex]
[tex]\[ t = \frac{(1234 - 1215)}{7.610519036176179} = 2.496544573331274 \][/tex]
5. Determine Degrees of Freedom:
Degrees of freedom (df) for the test are calculated using the formula:
[tex]\[ df = \frac{\left(\frac{sd_1^2}{n_1} + \frac{sd_2^2}{n_2}\right)^2}{\left(\frac{\left(\frac{sd_1^2}{n_1}\right)^2}{n_1 - 1} + \frac{\left(\frac{sd_2^2}{n_2}\right)^2}{n_2 - 1}\right)} \][/tex]
[tex]\[ df = \frac{\left(\frac{36^2}{50} + \frac{40^2}{50}\right)^2}{\left(\frac{\left(\frac{36^2}{50}\right)^2}{49} + \frac{\left(\frac{40^2}{50}\right)^2}{49}\right)} = 96.93188817100418 \][/tex]
6. Determine the Critical Value:
For a one-tailed test at [tex]\(\alpha = 0.05\)[/tex] significance level and [tex]\(df = 96.93\)[/tex], the critical value from the t-distribution table is approximately 1.6607.
7. Decision Rule:
Compare the test statistic to the critical value:
- If [tex]\(t > t_{\text{critical}}\)[/tex], we reject the null hypothesis.
Here, [tex]\(t = 2.4965\)[/tex] and [tex]\(t_{\text{critical}} = 1.6607\)[/tex].
Since [tex]\(2.4965 > 1.6607\)[/tex], we reject the null hypothesis.
8. Conclusion:
Since the test statistic is greater than the critical value, we reject the null hypothesis. This means that there is sufficient evidence at the 5% level of significance to conclude that the mean length of life for Type I bulbs is significantly greater than that for Type II bulbs. Therefore, Type I bulbs are superior to Type II bulbs concerning their length of life.
Here are the steps for this hypothesis test:
1. State the Hypotheses:
- Null Hypothesis [tex]\(H_0\)[/tex]: The mean length of life of Type I bulbs is equal to the mean length of life of Type II bulbs. [tex]\(\mu_1 = \mu_2\)[/tex]
- Alternative Hypothesis [tex]\(H_a\)[/tex]: The mean length of life of Type I bulbs is greater than the mean length of life of Type II bulbs. [tex]\(\mu_1 > \mu_2\)[/tex]
2. Significance Level:
- The significance level [tex]\(\alpha\)[/tex] is 0.05.
3. Calculate the Standard Error (SE):
The standard error of the difference in means is calculated using the formula:
[tex]\[ SE = \sqrt{\left(\frac{sd_1^2}{n_1}\right) + \left(\frac{sd_2^2}{n_2}\right)} \][/tex]
Using the given data:
- [tex]\(sd_1 = 36\)[/tex]
- [tex]\(n_1 = 50\)[/tex]
- [tex]\(sd_2 = 40\)[/tex]
- [tex]\(n_2 = 50\)[/tex]
[tex]\[ SE = \sqrt{\left(\frac{36^2}{50}\right) + \left(\frac{40^2}{50}\right)} = 7.610519036176179 \][/tex]
4. Calculate the Test Statistic:
The test statistic (t) is calculated using the formula:
[tex]\[ t = \frac{(\bar{x}_1 - \bar{x}_2)}{SE} \][/tex]
Where,
- [tex]\(\bar{x}_1 = 1234\)[/tex]
- [tex]\(\bar{x}_2 = 1215\)[/tex]
[tex]\[ t = \frac{(1234 - 1215)}{7.610519036176179} = 2.496544573331274 \][/tex]
5. Determine Degrees of Freedom:
Degrees of freedom (df) for the test are calculated using the formula:
[tex]\[ df = \frac{\left(\frac{sd_1^2}{n_1} + \frac{sd_2^2}{n_2}\right)^2}{\left(\frac{\left(\frac{sd_1^2}{n_1}\right)^2}{n_1 - 1} + \frac{\left(\frac{sd_2^2}{n_2}\right)^2}{n_2 - 1}\right)} \][/tex]
[tex]\[ df = \frac{\left(\frac{36^2}{50} + \frac{40^2}{50}\right)^2}{\left(\frac{\left(\frac{36^2}{50}\right)^2}{49} + \frac{\left(\frac{40^2}{50}\right)^2}{49}\right)} = 96.93188817100418 \][/tex]
6. Determine the Critical Value:
For a one-tailed test at [tex]\(\alpha = 0.05\)[/tex] significance level and [tex]\(df = 96.93\)[/tex], the critical value from the t-distribution table is approximately 1.6607.
7. Decision Rule:
Compare the test statistic to the critical value:
- If [tex]\(t > t_{\text{critical}}\)[/tex], we reject the null hypothesis.
Here, [tex]\(t = 2.4965\)[/tex] and [tex]\(t_{\text{critical}} = 1.6607\)[/tex].
Since [tex]\(2.4965 > 1.6607\)[/tex], we reject the null hypothesis.
8. Conclusion:
Since the test statistic is greater than the critical value, we reject the null hypothesis. This means that there is sufficient evidence at the 5% level of significance to conclude that the mean length of life for Type I bulbs is significantly greater than that for Type II bulbs. Therefore, Type I bulbs are superior to Type II bulbs concerning their length of life.
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