Question

There is no way for me to separate this question so please do all parts.

Explain why each of the confidence intervals were provided their respective labels. Break it up into 7 parts and use a, b, etc., subitem labeling

Reference:

	Write out example null and alternative hypotheses.	Sampling distribution used by test
One sample test for a mean	H0: mu = 0 H1: mu ≠ 0 (You may choose to write in words)	z-distribution if true standard deviation of population is known t-distribution if true standard deviation of population is unknown
Two sample tests for difference in means	Independent groups: H0: mu = mu2 H1: mu1 ≠ mu2	Two samples tests for difference in means focuses on comparing means of 2 independent groups using Z test for means and t test.
One sample test for a proportion	H0: p = p0 H1: p ≠ p0	One sample test for proportion tests if the proportion equates to a hypothesized value p0it uses Z test for single proportion.
Two sample tests for difference in proportions	H0: p1 = p2 H1: p1 ≠ p2	Two sample tests dior difference in proportions are used to contrast proportions of 2 independent groups that utilizes 2 sample Z test for proportion.
Test for paired difference	Paired Groups: H0: Mu d = 0 H1: Mu d ≠ 0	Test for paired difference are used to contrast the means of 2 paired groups that utilizes paired t test for means. Mu d = mean difference
Goodness-of- fit chi- squared test	H0: Oi = Ei Ha: Oi ≠ Ei	This is for testing if the observed distribution is what you expected. Oi = Observed Value . Ei= Expected Value
Chi-squared test for independenc e	H0: The 2 variables are not related/ associated. H1: The 2 variables are associated significantly.	The main point is to understand whether categorical/ nominal variables are related/ associated.
One-factor ANOVA	H0: mu1 = mu 2 = …. = mun H1: Means are not equal	The one factor ANOVA tests the effect of an independent factor with a specific number of levels over a dependent factor. A good example is comparing treatments. It’s important to recognize that if the treatment groups have means that are equal; it tells us that the dependent variable observations are stagnant while they are given the treatment. The treatment in this case has no effect that is significant.
Two-factor ANOVA	H0A= A has no effect H0B: B has no effect H0AB: There isn’t effect in the interaction H1A: A has an effect that is significant H1B: B has effect that is significant H1AB: There is a significant interaction effect	The main purpose of the two-factor ANOVA is to test whether two factors that are categorical with specific number of levels affect the continuous dependent variable. It also checks if their interaction causes an effect on dependent variable.
ANCOVA	H0: The adjusted population means have no difference. H1: The adjust population means have a significant difference	ANCOVA is for testing the effect of a factor that is independent with specific number levels over a factor that is dependent, whose effect we do not care about. ANCOVA is branch of ANOVA where it gives a way of having control of the linear effect of variables were not testing in the study regarded as covariates or control variables that is measured on an interval.
Pearson correlation (r)	H0: p = 0 H1: p ≠ 0

Simple linear regression (slope)	H0: Bi = 0, i = 1,2,.., n H1: 1 at least. Bi ≠ 0, i = 2,.., n
Mann- Whitney U	H0: Two groups have equal distribution of scores H1: Two groups have distribution of scores that aren’t equal.	Mann-Whitney U is utilized as a non-parametric alternate for Independent T test but there are no distributional assumptions are made unlike in T test where we assume that the population is distributed normally.
Sign test	H0: The medians of 2 groups equal to each other H1: The medians of the 2 groups do not equal each other.	Sign test is used as a Non-parametric substitution for Paired t test. In this case there isn’t any distributional assumptions done.
Wilcoxon signed-rank test	H0: Median equates with the value hypothesized H1: Median is significantly different from value hyptohesized.	Wilcoxon siged-rank test is for comparing the median against the value hypothesized.
Kruskal- Wallis test	H0: The groups have the same mean ranks H1: The groups do not have the same mean ranks	This is utilized as a Non-parametric alternative for One way ANOVA but in this case there isn’t any distributional assumptions made.
Spearman rank correlation coefficient	H0: p = 0 H1: p ≠ 0	This is the Non-parametric alternative for Pearson’s correlation coefficient r, to gauge the direction of association and as well as the strength between two ranked variables but the key is that there are no assumptions on the linearity of the relationship done
Logistic regression	H0: Bi = 0, i = 1,2,..., n H1: One at least Bi ≠, 0,i = 1,2,..., n	Logistic regressions tests if a significant causal relationship between the dependent variable measured in normal scale and the independent variable. We see instead of predicting the dependent variable, the probabilities are predicted.
Log-rank test	H0: No difference in survival between 2 or more independent groups H1: The difference is significant in survival between 2 or more independent groups	This is comparing distribution of survival of 2 or more independent groups
Cox proportional hazards regression	H0: Bi = 0, i = 1, 2,..n H1: One at least Bi ≠ 0, i = 1,2.. n

Answer 1

Solution :

One sample test for a mean	H0: = 0 H1: ≠ 0	z-distribution if true standard deviation of population is known t-distribution if true standard deviation of population is unknown
Two sample tests for difference in means	Independent Groups: H0: H1:	- Used to compare the means of two independent groups using Z test for means (if true standard deviation of population is known) and t test (if true standard deviation of population is unknown)
One sample test for a proportion		Used to test whether the proportion is equal to a hypothesized value p0 ,using Z test for single proportion
Two sample tests for difference in proportions		Used to compare the proportions of two independent groups using two sample Z test for proportion
Test for paired difference	Paired Groups:	- Used to compare the means of two dependent / paired groups using Paired t test for means di = Difference of observations, is the mean difference
Goodness-of- fit chi- squared test		Used to test whether the observed distribution is same as expected Oi, Ei = Observed and Expected values respectively
Chi-squared test for independence	H0: There is no association between the two variables Ha: There is a significant association between the two variables	- Used to test whether two nominal / categorical variables are associated.
One-factor ANOVA	Ha: Not all means are equal	Used to test the effect of an independent variable (factor) with certain no. levels (n) over a dependent variable i.e. to compare 'n' treatments: If treatment group means equal - it would imply that the dependent variable observations remains the same irrespective of the treatment administered. i.e treatment has no significant effect.
Two-factor ANOVA	H0a: Factor A has no effect H0b: Factor B has no effect H0ab: There is no interaction effect H1a: Factor A has a significant effect H1b: Factor B has a significant effect H1ab: There is a significant interaction effect	Used to test whether two factors (categorical) with certain no. of levels affect the (continuous) dependent variable. And whether their interaction has any effect on dependent variable.
ANCOVA	H0: There is no difference among the adjusted population means H0: There is a significant difference among the adjusted population means	Used to test the effect of an independent variable (factor) with certain no. levels (n) over a dependent variable, controlling for the variable, whose effect we are not interested in i.e. to compare 'n' treatments. It is an extension of ANOVA that provides a way of statistically controlling the (linear) effect of variables we do not want to examine in the study, called covariates, or control variables measured on an interval or ratio scale.
Pearson correlation (r)		Used to test whether a significant linear relationship between two continuous variables,and its the strength and direction. - Pearson's Linear Correlation coefficient

Simple linear regression (slope)	At least one	Used to test whether a significant linear causal relationship between the response (dependent variable) measured in interval / ratio scale and the ith predictor (independent variable) in the regression model
Mann- Whitney U	H0: The distribution of scores for the two groups are equal Ha: The distribution of scores (Mean rank) for the two groups are not equal	Used as a Non - parametric alternative for Independent t test, where, no distributional assumprions is made (unlike in t test where we assume that the population is normally distributed)
Sign test	H0:Medians of the two groups are equal Ha: Medians of the two groups are not equal	Used as a Non - parametric alternative for Paired t test, where, no distributional assumprions is made (unlike in paired t test where we assume that the population is normally distributed)
Wilcoxon signed-rank test	H0:Median is equal to hypothesized value Ha: Median is significantly different from hypothesized value	Used to compare the median against a hypothesized value
Kruskal- Wallis test	H0: The mean ranks of all the n groups are same Ha: The mean ranks of the n groups are not the same	Used as a Non - parametric alternative for One way ANOVA, where, no distributional assumprions is made (unlike in One way ANOVA where we assume that the population is normally distributed and that the variances are homogenous)
Spearman rank correlation coefficient		Used as a Non - parametric alternative for Pearson's correlation coefficient r, to measure the strength and direction of association between two ranked variables where, no assumptions on the linearity of the relationship is made (unlike in Pearson's r where we assume that the relationship is linear)
Logistic regression	At least one	Used to test whether a significant causal relationship between the response (dependent variable) measured in nominal scale and the ith predictor (independent variable) in the regression model. Here, instead of predicting the dependent variable itself (As in linear regression) we predict the probabilities.
Log-rank test	H0: There is no difference in survival between two or more independent groups Ha: There is a significant difference in survival between two or more independent groups	Used to compare the survival distribution of two or more independent groups
Cox proportional hazards regression	At least one	Used to test whether a significant causal relationship between the response Survival time (dependent variable) with one or more predictors in the regression model.

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