An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (2)
To find the number of electrons flowing through a device delivering a current of 15.0 A for 30 seconds, we calculate the total charge, which is 450 C. Dividing this charge by the charge of a single electron (1.6 × 10^-19 C) gives us about 2.81 × 10^21 electrons. Hence, approximately 2.81 × 10^21 electrons flow through the device in that time.
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Calculate the sample means and standard deviations for both white bread with crust and wheat bread without crust.
Compute the t-statistic to compare the means of the two samples: t = 0.266 .
Find the critical t-value for a one-tailed test with α = 0.025 and df = 20 : t α = 2.086 .
Since t < t α , fail to reject the null hypothesis. There is not enough evidence to conclude that white bread with the crust has more fiber than wheat bread without the crust. False
Explanation
Understand the problem and provided data We are given data from a study comparing the fiber content of white bread with the crust versus wheat bread without the crust. We want to determine if there is a statistically significant difference in fiber content between white bread with the crust and wheat bread without the crust, specifically testing if white bread with the crust has more fiber. The significance level is α = 0.025 .
Define the null and alternative hypotheses The null hypothesis H 0 is that the mean fiber content of white bread with crust is less than or equal to the mean fiber content of wheat bread without crust: μ 1 ≤ μ 2 .
The alternative hypothesis H 1 is that the mean fiber content of white bread with crust is greater than the mean fiber content of wheat bread without crust: \mu_2"> μ 1 > μ 2 . This is a right-tailed test.
Calculate sample means and standard deviations We calculate the sample mean and sample standard deviation for both samples. For white bread with crust, the sample mean is x 1 ˉ = 1.059 and the sample standard deviation is s 1 = 0.135 . For wheat bread without crust, the sample mean is x 2 ˉ = 1.037 and the sample standard deviation is s 2 = 0.235 .
Calculate the t-statistic We calculate the t-statistic using the formula: t = n 1 s 1 2 + n 2 s 2 2 x 1 ˉ − x 2 ˉ where n 1 = n 2 = 11 . Plugging in the values, we get: t = 11 0.13 5 2 + 11 0.23 5 2 1.059 − 1.037 = 11 0.018225 + 11 0.055225 0.022 = 0.001657 + 0.005020 0.022 = 0.006677 0.022 = 0.0817 0.022 = 0.269 The t-statistic is approximately 0.269 .
Determine the degrees of freedom and critical t-value The degrees of freedom are df = n 1 + n 2 − 2 = 11 + 11 − 2 = 20 . The critical t-value for a one-tailed test with α = 0.025 and df = 20 is t α = 2.086 .
Compare the t-statistic with the critical t-value Since the calculated t-statistic ( 0.266 ) is less than the critical t-value ( 2.086 ), we fail to reject the null hypothesis.
State the conclusion Conclusion: There is not enough evidence at the 2.5% significance level to conclude that there is more fiber in white bread with the crust than in wheat bread without the crust.
Examples
This type of hypothesis testing can be used in nutritional studies to compare the health benefits of different food products. For example, a researcher might want to compare the effectiveness of a new weight loss supplement to a placebo. By collecting data on weight loss for both groups and performing a hypothesis test, the researcher can determine if the supplement has a statistically significant effect.
