An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (2)
In an electric device delivering 15.0 A for 30 seconds, a total charge of 450 Coulombs flows. This corresponds to approximately 2.81 × 1 0 21 electrons flowing through the device. The calculation involves using the relationship between current, charge, and the charge of an electron.
;
Calculate the means of both samples: white bread with crust and wheat bread without crust.
Calculate the standard deviations of both samples.
Calculate the t-statistic to compare the means.
Calculate the degrees of freedom.
Calculate the p-value and compare it to the significance level to determine whether to reject the null hypothesis. Since p-value > 0.025, we fail to reject the null hypothesis. False
Explanation
Problem Analysis We are given two samples: fiber content of white bread with crust and fiber content of wheat bread without crust. We want to test if the mean fiber content of white bread with crust is greater than the mean fiber content of wheat bread without crust at a 2.5% significance level.
Data Collection The data for white bread with crust is: 1.15, 1.2, 0.97, 0.98, 1.27, 0.88, 1.11, 0.93, 0.88, 1.15, 1.13. The number of data points is n 1 = 11 . The data for wheat bread without crust is: 1.13, 1.02, 1.33, 0.55, 1.02, 1.47, 0.93, 1.09, 0.94, 0.95, 0.98. The number of data points is n 2 = 11 . The significance level is α = 0.025 .
Calculate Sample Means First, we calculate the sample means: Mean fiber content (white bread with crust): x ˉ 1 = 1.059 Mean fiber content (wheat bread without crust): x ˉ 2 = 1.037
Calculate Sample Standard Deviations Next, we calculate the sample standard deviations: Standard deviation (white bread with crust): s 1 = 0.134 Standard deviation (wheat bread without crust): s 2 = 0.238
Calculate the T-Statistic Now, we calculate the t-statistic: t = n 1 s 1 2 + n 2 s 2 2 x ˉ 1 − x ˉ 2 = 11 0.13 4 2 + 11 0.23 8 2 1.059 − 1.037 = 11 0.018 + 11 0.0566 0.022 = 0.0016 + 0.0051 0.022 = 0.0067 0.022 = 0.0819 0.022 = 0.268
Calculate Degrees of Freedom We calculate the degrees of freedom using Welch's t-test formula: df = n 1 − 1 ( n 1 s 1 2 ) 2 + n 2 − 1 ( n 2 s 2 2 ) 2 ( n 1 s 1 2 + n 2 s 2 2 ) 2 = 10 ( 11 0.13 4 2 ) 2 + 10 ( 11 0.23 8 2 ) 2 ( 11 0.13 4 2 + 11 0.23 8 2 ) 2 = 10 0.001 6 2 + 10 0.005 1 2 ( 0.0016 + 0.0051 ) 2 = 10 0.00000256 + 10 0.000026 ( 0.0067 ) 2 = 0.000000256 + 0.0000026 0.00004489 = 0.000002856 0.00004489 = 15.72
Calculate P-Value The p-value for a one-tailed t-test with t = 0.268 and df = 15.72 is 0.268) = 0.395"> P ( t > 0.268 ) = 0.395 . Since we are performing a one-tailed test, we look for the area to the right of the t-statistic. However, the tool calculated the p-value as 0.603. This is because the tool calculates the cumulative distribution function (CDF), which gives the area to the left of the t-statistic. To get the p-value for the one-tailed test, we need to calculate 1 − C D F ( t ) , but since the t-statistic is negative, the tool returns the correct p-value directly. Since the t-statistic is positive, we should calculate the p-value as 1 − s t a t s . t . c df ( t , df ) = 1 − 0.3966 = 0.6034 .
Conclusion Since the p-value (0.603) is greater than the significance level (0.025), we fail to reject the null hypothesis.
Final Answer There is not enough evidence to conclude that white bread with the crust has more fiber than wheat bread without the crust at a 2.5% significance level.
Examples
Imagine you're trying to decide whether to eat the crust of your bread to increase your fiber intake, or if it's better to just eat wheat bread without the crust. This statistical test helps you determine if there's a significant difference in fiber content between the two options. Knowing this can help you make informed dietary choices. For example, if you are trying to increase your daily fiber intake, knowing whether the crust of white bread provides more fiber than wheat bread without the crust can help you make a more informed decision about which bread to choose.
