This document provides definitions and steps for hypothesis testing of population means and proportions. It defines key terms like null and alternative hypotheses, type 1 and type 2 errors, level of significance, test statistics, and p-values. It also outlines the steps for hypothesis testing which include formulating hypotheses, collecting data, computing test statistics, determining critical regions, and making conclusions. Several examples of hypothesis tests on population means and proportions are provided to illustrate the concepts.
This document provides definitions and steps for hypothesis testing of population means and proportions. It defines key terms like null and alternative hypotheses, type 1 and type 2 errors, level of significance, test statistics, and p-values. It also outlines the steps for hypothesis testing which include formulating hypotheses, collecting data, computing test statistics, determining critical regions, and making conclusions. Several examples of hypothesis tests on population means and proportions are provided to illustrate the concepts.
This document provides definitions and steps for hypothesis testing of population means and proportions. It defines key terms like null and alternative hypotheses, type 1 and type 2 errors, level of significance, test statistics, and p-values. It also outlines the steps for hypothesis testing which include formulating hypotheses, collecting data, computing test statistics, determining critical regions, and making conclusions. Several examples of hypothesis tests on population means and proportions are provided to illustrate the concepts.
This document provides definitions and steps for hypothesis testing of population means and proportions. It defines key terms like null and alternative hypotheses, type 1 and type 2 errors, level of significance, test statistics, and p-values. It also outlines the steps for hypothesis testing which include formulating hypotheses, collecting data, computing test statistics, determining critical regions, and making conclusions. Several examples of hypothesis tests on population means and proportions are provided to illustrate the concepts.
Population Mean Definitions •A null hypothesis is a statement of equality or no difference.
•An alternative hypothesis is an opposing
statement believed to be true whenever the null hypothesis is rejected. Definitions •A directional alternative hypothesis involves quantifier < or >.
•A non-directional alternative hypothesis involves
the quantifier “not equal to”. Definitions •A statistical test of hypothesis is a method or tool used to decide whether or not to reject a statistical hypothesis. Definitions •A one-tailed test is used to test a null hypothesis against a directional alternative hypothesis.
•A two-tailed test is used to test a null hypothesis
against a non-directional alternative hypothesis. Definitions •Type 1 error is an error of rejecting a null hypothesis when in fact it is true.
•Type 2 error is an error of accepting a null
hypothesis when in fact it is false. Consequences of Hypothesis Testing
Decision Null Hypothesis Null Hypothesis
True False Do not reject Correct Type 2 error Null Hypothesis Decision Reject Null Type 1 Error Correct Hypothesis Decision Definitions •Level of significance is the probability of committing a type 1 error. •A test statistic is a numerical value computed from the sample data. •A significance probability or p-value is the lowest level of significance at which the test statistic value is significant. Steps 1. Formulate the null and alternative hypothesis. 2. Collect data and decide on an appropriate statistical testing procedure. 3. Compute the test-statistic or the probability value (p- value). 4. Determine the critical region, also called the rejection region. 5. Make a decision and a conclusion about the hypotheses. Summarizing Results (Bluman, 2014) 𝜎 is known Automotive engineers tested the gas mileage (in kilometers per liter or km/L) of a passenger car model from a certain car company. A random sample of 35 cars resulted to a mean gas mileage of 15km/L and a standard deviation of 2.5 km/L. The car company claims that the passenger car model has an average gas mileage of 16 km/L. Test if the claim is valid at 5% level of significance. Exercise A factory manufacturing light-emitting diode (LED) bulbs claims that their light bulbs last for 50 000 hours on the average. To confirm if this claim is valid, a quality control manager got a sample of 50 LED bulbs and obtained a mean lifespan of 40 000 hours. The standard deviation of the manufacturing process is 1000 hours. Do you think that the claim of the manufacturer is valid at the 5% level of significance? Exercise The Mathematics Department in a certain university is conducting a study to determine how long it takes its graduates to find a job. A sample of 36 graduates was surveyed and it was found that the average time it has taken a graduate to find a new job is 3.5 months, with a standard deviation of 1.5 months. Is there sufficient evidence to conclude that the graduates of this department take on the average more than three months to find a job at 10% level of significance? Exercise A manufacturer produces paper that has a mean length of 11 in. and a standard deviation of 0.02 in. The 20 sheets sampled have a mean paper length of 10.98 in. Assuming that the lengths of the produced papers are normally distributed, can you conclude that the mean length of papers produced by this company is less than 11 inches? Use 1% level of significance. 𝜎 is unknown Pauline heard that the average grade in mathematics of her class is at least 88%. She was not convinced by this, and so decided to use hypothesis testing to check if this claim was true. She got a random sample of 10 classmates who gave their grades in mathematics as follows: 90, 93, 85, 77, 88, 80, 78, 83, 95, 90 Assume that the distribution of the grades is normal. Based on this sample data, what would Pauline’s conclusion be on the average grade in mathematics of her class at 5% level of significance? Exercise When certain air pollutants react with rainwater, acid rain that corrodes exposed metals is produced. Suppose that water samples from 8 instances of rainfall are collected and analyzed for power of hydrogen (pH) levels: 3.7, 3.9, 4.0, 3.5, 4.2, 4.5, 4.1, 3.8 Do you think that there is a reason to believe that the pH of rainwater is now greater that 3.5? Assume that the pH level of water samples are normally distributed. Use 5% level of significance. Exercise Physicians say that the normal temperature of a person’s body is 37 degrees Celsius. In a class, the temperatures (in degrees Celsius) of 10 students were collected. These were: 36.8, 37.3, 36.5, 37, 37.5, 37.1, 36.7, 37, 37.1, 36.9 Test the hypothesis that the students in this class have normal body temperatures. Assume that body temperature is normally distributed. Use 1% level of significance. Population Proportion A survey is conducted to determine the opinions of people on global warming. In a random sample of 150 people, 108 think that global warming is a serious world problem. Is there sufficient evidence that the proportion of people who regard global warming as a serious problem is significantly higher than 60%? Use 1% level of significance. Exercise A presidential candidate asks a polling organization to conduct a nationwide survey to determine the percentage of potential voters who would vote for him over his rival presidential candidate. Out of 2500 respondents in the sample, 925 said they would vote for him. If 40% of the potential voters vote for his rival, is this significantly different from the percentage of potential voters of the candidate who requested the survey? Use 5% level of significance. Exercise In a survey on TV show ratings, 1000 viewers were asked if they regularly watch a certain singing competition. Of the 1000 surveyed, 338 answered “Yes”. The network considers canceling the show if less than one-third of the population regularly watch it. Can you help the network decide using hypothesis testing at 10% level of significance? Exercise Beefy Burger, a fast-food restaurant claims that 85% of the burger fanatics prefer to eat in their place. To test the claim, a random sample of 90 burger customers are selected at random and asked what they prefer. If 76 of the 90 burger fanatics said they prefer to eat at Beefy Burger, what conclusion do we draw? Use a 0.05 level of significance. Exercise Haus of Gaz claims that more than two-thirds of the houses in a certain subdivision use their brand. Do we have reason to doubt this claim if in a random sample of 40 houses in this subdivision, it is found that 25 use the company’s brand. Use a 0.01 level of significance. Exercise A congressman is hoping that his bill is favored by his constituents to increase his chance of getting reelected in the next election. He asked his research office to conduct a survey on this matter to verify whether or not he can get support from his constituents so that if he gets a 90% support for the bill, he would certainly start with his campaign. A random sample of 150 respondents yielded 128 who are in favor of his bill. Test the hypothesis that p=0.9 against the alternative p<0.9 at 0.05 level of significance. Exercise MED Drug Company designed a new drug to prevent colds. The company states that this drug is about 95% effective. To test this claim, they chose a sample of 200 subjects and found out that 15 of these subjects caught a cold. Based on these findings, can we reject the company’s claim that the drug is 95% effective? Use a 0.01 level of significance.