Statistics

In Statistics, there are 2 main activities of inferential  statistics that are using sample data to

1. estimate a population parameter

2. test a hypothesis about a population parameter

A hypothesis test is a standard procedure for testing a claim about to parameter of a population.

The null hypothesis is a statement that the value of a population parameter is:

• Equal to =a specific value 

• Larger or equal to a specific value

• Smaller or equal to a specific value

Components of Hypothesis Test

the alternative hypothesis is a statement that the parameter has a value that somehow differ from the null hypothesis 

• An alternative hypothesis is a complimentary event of a null hypothesis

• the symbolic form of alternative hypothesis must use ≠,<,>.

with  the defined null and alternative hypothesis

• we will test the null hypothesis

• Either reject H₀ or fail to reject H₀ .

Example

Consider the claim that mean weight of airline passengers at most 195ib. Identify the null hypothesis and alternative hypothesis.

Answer

Step 1 Express the given claim in symbolic form. The mean is at most 195ib which is µ≤195.

Step 2 if µ≤195 is false , then µ>195 must be true.

Step 3 symbolic expressions 

µ≤195 and µ>195 

µ>195  does not contain equality

H₀ andH₁ is defined based on a claim

• Significance level of a test the probability that the test statistic will fall in the critical region when the null hypothesis is true. If the  test statistic falls in the critical region, we reject the null hypothesis, so is the probability of making the mistake of rejecting the null hypothesis when it true.

• A random sample to produce a test statistics to decide of either rejecting  or failing to reject H₀

The test statistic is a value used in deciding about the null hypothesis and is found by converting the sample statistic to a  score with assumption that the null hypothesis is true.

Critical region and Value

The critical region is the set of all values of test statistic that cause to reject null hypothesis. The critical region is determined by significance level for test. A critical value is the value that separates the critical region from values of test statistic that do not lead to rejection of the null hypothesis. The critical value depends on the nature of the null hypothesis, sampling distribution that applies, and the significance level α.

Types of Hypothesis Tests: Two=tailed, left and right -tailed

the tails in a distribution are extreme regions bounded by critical. The critical region and non critical region might be separated by using two tails or either the left tail or right tail only. it therefore becomes important to correctly characterize a hypothesis test as two- tailed, left-tailed or right- tailed.

A manufacturer is filling 40 oz. packages with flour. The company wants the package contents to average 40 oz.

H₀ µ =40

H₀ µ ≠40

The P-value (or p-value or probability value) is the probability of getting a value of the test statistic that is at least as extreme as the one representing the sample data, assuming that the null hypothesis is true.

Critical region in the left tail: P-value = area to the left of the test statistic
Critical region in the right tail: P-value = area to the right of the test statistic
Critical region in two tails: P-value = twice the area in the tail beyond the test statistic

The null hypothesis is rejected if the P-value is very small
Here is a memory tool useful for interpreting the P-value: If the P-value is low, the null must go.(The risk is that you’re willing to take of making a wrong decision—then you reject the null hypothesis)
If the P-value is high, the null will fly. (If the p is high, the null will fly and fail to reject the null hypothesis.
 

Decision Criteria

1. P-value method: using the significance level α

   if P-value ≤α, reject H₀ 

   if P-value >α,fail to reject H₀ 

2. Traditional Method

   if the test statistic falls within the critical region, reject H₀ 

   if the test statistic does not fall within the critical region, fail to reject H_0 .

Errors in Hypothesis Tests

Type I Error

A type I error is the mistake of rejecting the null hypothesis when it is actually true. the symbol α is used to represent the probability of type I error.

Type II Error

A type II error is the mistake of accepting the null hypothesis when it is actually false. The symbol for type II error is β.

Example:

Assume that we are conducting a hypothesis test of claim that a method of gender selection increases the likelihood of baby girl, so that the probability of  a baby girl is p>0.5. here are the null and alternative hypotheses:

H₀ =0.5 and   H_a >0.5 

A type I error is the mistake of rejecting a true null hypothesis, so this is a type I error: Conclude that there is sufficient evidence to support p>0.5 when in reality p=0.5.

A type II error is the mistake of failing to reject the null hypothesis when it is false, so this is a  type II error: Fail to reject p=0.5 (and therefore fail to support p>0.5) when in reality p>0.5.
 

Testing Claims  About a Population Proportion (p)

Requirements for Testing Claims About a Population Proportion p
The sample
observations are a simple random sample.
2) The conditions for a binomial distribution are satisfied.
3) The conditions np ≥ 5 and nq≥ 5 are both satisfied, so the binomial distribution of sample  
proportions can be approximated by a normal distribution with  µ = np and  𝜎= n.p.q

 P-values: Use the standard normal distribution - Critical Values: Use the standard normal distribution

The text refers to a study in which 57 out of 104 pregnant women correctly guessed  the sex of their babies. Use these sample data to test the claim that the success rate of such guesses is no different from the 50% success rate expected with random chance guesses. Use a 0.05 significance level.

Testing Claims About a Mean: 𝜎 not Known