A random sample of n= 4 individuals is selected from a population with μ = 35, and a treatment is administered to each individual in the sample. After treatment, the sample mean is found to be M = 40.1 with SS = 48.How much difference is there between the mean for the treated sample and the mean for the original population? (Note: In a hypothesis test, this value forms the numerator of the t statistic.)If there is no treatment effect, how much difference is expected between the sample mean and its population mean? That is, find the standard error for M. (Note: In a hypothesis test, this value is the denominator of the t statistic.)Based on the sample data, does the treatment have a significant effect? Use a two-tailed test with α = .05.

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Description: A random sample of n= 4 individuals is selected from a population with μ = 35, and a treatment is administered to each individual in the sample. After treatment, the sample mean is found to be M = 40.1 with SS = 48.How much difference is there betwe

a. A sample of 4 individuals is selected from a population with mean 35. The sample mean and sum of squared difference (SS) is found to be 40.1 and 48 respectively. Calculations: The difference can be calculated using formula: M−μ=40.1−35=5.1 Conclusion: Hence, the difference is 5.1.b. Calculate sample standard deviation, s, using formula: s=√(SS/n−1) Substitute SS=48 and n=4 in the formula s=√(48/4−1)= √(48/3)=4 Now, formula for estimated sample error, sm, is: sm=s/√n Substitute s=4 and n=4 in the formula sm =4/√4=4/2=2 Conclusion: Hence, the standard error is 2.c. A sample of 4 individuals is selected from a population with mean 35. The sample mean and sum of squared difference (SS) is found to be 40.1 and 48 respectively. The standard deviation is 4. Two-tailed hypothesis test using α=0.05 is performed. Calculations: Step 1: Null Hypothesis is H0:μ=35 and Alternate Hypothesis is H1:μ≠35. Step 2: For a sample of n=4, the t-statistics will have (n−1) degrees of freedom, i.e. df=3. For a two-tailed test with α=0.05 and df=3, the critical value (CV) is obtained from the t-table as t=±3.182. Step 3: t-statistics can be calculated using formula: t=(M−μ)/sm Substitute 40.1 for M, 35 for μ and 2 for sm in the formula t=(40.1−35)/2=5.1/2=2.55 Step 4: Rejection rule: Reject when t−statistics<−3.182 or t−statistics>3.182 Since t−statistics(=2.55)<critical-value(=3.182), we fail to reject the null hypothesis. Step 5: Based on the results of hypothesis test, we don’t have sufficient evidence to reject the null hypothesis. Conclusion: Hence, treatment doesn’t have a significant effect.

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MATLAB: An Introduction with Applications

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ISBN:9781119256830

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A random sample of n= 4 individuals is selected from a population with μ = 35, and a treatment is administered to each individual in the sample. After treatment, the sample mean is found to be M = 40.1 with SS = 48.

How much difference is there between the mean for the treated sample and the mean for the original population? (Note: In a hypothesis test, this value forms the numerator of the t statistic.)
If there is no treatment effect, how much difference is expected between the sample mean and its population mean? That is, find the standard error for M. (Note: In a hypothesis test, this value is the denominator of the t statistic.)
Based on the sample data, does the treatment have a significant effect? Use a two-tailed test with α = .05.

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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