Question

Describe in 175 words please type response:

Describe statistical inference and how it corralates with hypothesis testing for single populations.

Describe in 175 words please type response:

Describe how decision making is done using one sample hypothesis testing.

Answer 1

Describe statistical inference and how it corralates with hypothesis testing for single populations

Statistical inference is the process of using data analysis to deduce properties of an underlying probability distribution. Inferential statistical analysis infers properties of a population, for example by testing hypotheses and deriving estimates. ... Inferential statistics can be contrasted with descriptive statistics.Descriptive statistics is solely concerned with properties of the observed data, and it does not rest on the assumption that the data come from a larger population.

Hypothesis testing or significance testing is a method for testing a claim or hypothesis about a parameter in a population, using data measured in a sample. In this method, we test some hypothesis by determining the likelihood that a sample statistic could have been selected, if the hypothesis regarding the population parameter were true.

Statistical hypothesis testing plays an important role in the whole of statistics and in statistical inference. For example, Lehmann (1992) in a review of the fundamental paper by Neyman and Pearson (1933) says: "Nevertheless, despite their shortcomings, the new paradigm formulated in the 1933 paper, and the many developments carried out within its framework continue to play a central role in both the theory and practice of statistics and can be expected to do so in the foreseeable future".

A test of a statistical hypothesis, where the region of rejection is on only one side of the sampling distribution, is called a one-tailed test(or testing for single population). For example, suppose the null hypothesis states that the mean is less than or equal to 10. The alternative hypothesis would be that the mean is greater than 10. The region of rejection would consist of a range of numbers located on the right side of sampling distribution; that is, a set of numbers greater than 10.​​​​​​​

Statistical inference is often based on a test of significance, “a procedure by which one determines the degree to which collected data are consistent with a specific hypothesis…” (Matthews and Farewell, 1996). Hypotheses may be specific, for example, that the slope relating two variables is 1, the line of identity. More often the hypothesis is that two (or more) sample statistics could have been drawn from the same population. The principles to be discussed are illustrated by referring to sample and population means, but apply equally to all other types of statistics. This is the null hypothesis or H₀, and if it is accepted then we believe that the two samples could have come from the same population. If we choose to reject the null hypothesis, then there is an alternative hypothesis, H_A .

Describe how decision making is done using one sample hypothesis testing

We use inferential statistics because it allows us to measure behavior in samples to learn more about the behavior in populations that are often too large or inaccessible. We use samples because we know how they are related to populations.​​​​​​​

The method in which we select samples to learn more about characteristics in a given population is called hypothesis testing. Hypothesis testing is really a systematic way to test claims or ideas about a group or population.​​​​​​​

When we know the mean and standard deviation in a single population, we can use the one–independent sample z test,t we can state one of three alternative hypotheses: A population mean is greater than (>), less than (<), or not equal (≠) to the value stated in a null hypothesis. The alternative hypothesis determines which tail of a sampling distribution to place the level of significance.

We will use four steps to illustrate hypothesis testing:

Step 1: State the hypotheses. The population mean is 558, and we are testing whether the null hypothesis is (=) or is not (≠) correct:

H_o : m = 558 Mean test scores are equal to 558 in the population.

H₁ : m ≠ 558 Mean test scores are not equal to 558 in the population.

Step 2: Set the criteria for a decision. The level of significance is .05, which makes the alpha level a = .05. To locate the probability of obtaining a sample mean from a given population, we use the standard normal distribution. We will locate the z scores in a standard normal distribution that are the cutoffs, or critical values, for sample mean values with less than a 5% probability of occurrence if the value stated in the null (m = 558) is true.

Step 3: Compute the test statistic. Step 2 sets the stage for making a decision because the criterion is set. The probability is less than 5% that we will obtain a sample mean that is at least 1.96 standard deviations above or below the value of the population mean stated in the null hypothesis. In this step, we will compute a test statistic to determine whether the sample mean we selected is beyond or within the critical values we stated in Step 2.

Step 4: Make a decision. To make a decision, we compare the obtained value to the critical values. We reject the null hypothesis if the obtained value exceeds a critical value. Figure 8.5 shows that the obtained value (Z_obt = 1.94) is less than the critical value; it does not fall in the rejection region. The decision is to retain the null hypothesis.