# 8.5 Additional information and full hypothesis test examples --  (Page 3/51)

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Statistics students believe that the mean score on the first statistics test is 65. A statistics instructor thinks the mean score is higher than 65. He samples ten statistics students and obtains the scores

• 65
• 65
• 70
• 67
• 66
• 63
• 63
• 68
• 72
• 71
. He performs a hypothesis test using a 5% level of significance. The data are assumed to be from a normal distribution.

Set up the hypothesis test:

A 5% level of significance means that α = 0.05. This is a test of a single population mean .

H 0 : μ = 65   H a : μ >65

Since the instructor thinks the average score is higher, use a ">". The ">" means the test is right-tailed.

Determine the distribution needed:

Random variable: $\overline{X}$ = average score on the first statistics test.

Distribution for the test: If you read the problem carefully, you will notice that there is no population standard deviation given . You are only given n = 10 sample data values. Notice also that the data come from a normal distribution. This means that the distribution for the test is a student's t .

Use t df . Therefore, the distribution for the test is t 9 where n = 10 and df = 10 - 1 = 9.

Calculate the p -value using the Student's t -distribution:

p -value = P ( $\overline{x}$ >67) = 0.0396 where the sample mean and sample standard deviation are calculated as 67 and 3.1972 from the data.

Interpretation of the p -value: If the null hypothesis is true, then there is a 0.0396 probability (3.96%) that the sample mean is 65 or more.

Compare α and the p -value:

Since α = 0.05 and p -value = 0.0396. α > p -value.

Make a decision: Since α > p -value, reject H 0 .

This means you reject μ = 65. In other words, you believe the average test score is more than 65.

Conclusion: At a 5% level of significance, the sample data show sufficient evidence that the mean (average) test score is more than 65, just as the math instructor thinks.

The p -value can easily be calculated.

## Try it

It is believed that a stock price for a particular company will grow at a rate of $5 per week with a standard deviation of$1. An investor believes the stock won’t grow as quickly. The changes in stock price is recorded for ten weeks and are as follows: $4,$3, $2,$3, $1,$7, $2,$1, $1,$2. Perform a hypothesis test using a 5% level of significance. State the null and alternative hypotheses, find the p -value, state your conclusion, and identify the Type I and Type II errors.

H 0 : μ = 5

H a : μ <5

p = 0.0082

Because p < α , we reject the null hypothesis. There is sufficient evidence to suggest that the stock price of the company grows at a rate less than $5 a week. Type I Error: To conclude that the stock price is growing slower than$5 a week when, in fact, the stock price is growing at $5 a week (reject the null hypothesis when the null hypothesis is true). Type II Error: To conclude that the stock price is growing at a rate of$5 a week when, in fact, the stock price is growing slower than $5 a week (do not reject the null hypothesis when the null hypothesis is false). Joon believes that 50% of first-time brides in the United States are younger than their grooms. She performs a hypothesis test to determine if the percentage is the same or different from 50% . Joon samples 100 first-time brides and 53 reply that they are younger than their grooms. For the hypothesis test, she uses a 1% level of significance. Set up the hypothesis test: The 1% level of significance means that α = 0.01. This is a test of a single population proportion . H 0 : p = 0.50 H a : p ≠ 0.50 The words "is the same or different from" tell you this is a two-tailed test. Calculate the distribution needed: Random variable: P′ = the percent of of first-time brides who are younger than their grooms. Distribution for the test: The problem contains no mention of a mean. The information is given in terms of percentages. Use the distribution for P′ , the estimated proportion. ${P}^{\prime }~N\left(p,\sqrt{\frac{p\cdot q}{n}}\right)$ Therefore, ${P}^{\prime }~N\left(0.5,\sqrt{\frac{0.5\cdot 0.5}{100}}\right)$ where p = 0.50, q = 1− p = 0.50, and n = 100 Calculate the p -value using the normal distribution for proportions: p -value = P ( p′ <0.47 or p′ >0.53) = 0.5485 where x = 53, p′ = = 0.53. Interpretation of the p -value: If the null hypothesis is true, there is 0.5485 probability (54.85%) that the sample (estimated) proportion $p\text{'}$ is 0.53 or more OR 0.47 or less (see the graph in [link] ). μ = p = 0.50 comes from H 0 , the null hypothesis. p′ = 0.53. Since the curve is symmetrical and the test is two-tailed, the p′ for the left tail is equal to 0.50 – 0.03 = 0.47 where μ = p = 0.50. (0.03 is the difference between 0.53 and 0.50.) Compare α and the p -value: Since α = 0.01 and p -value = 0.5485. α < p -value. Make a decision: Since α < p -value, you cannot reject H 0 . Conclusion: At the 1% level of significance, the sample data do not show sufficient evidence that the percentage of first-time brides who are younger than their grooms is different from 50%. The p -value can easily be calculated. The Type I and Type II errors are as follows: The Type I error is to conclude that the proportion of first-time brides who are younger than their grooms is different from 50% when, in fact, the proportion is actually 50%. (Reject the null hypothesis when the null hypothesis is true). The Type II error is there is not enough evidence to conclude that the proportion of first time brides who are younger than their grooms differs from 50% when, in fact, the proportion does differ from 50%. (Do not reject the null hypothesis when the null hypothesis is false.) #### Questions & Answers explain and give four Example hyperbolic function Lukman Reply The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu SABAL Reply 1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3 Pawel 2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31 Pawel ok Ifeanyi on number 2 question How did you got 2x +2 Ifeanyi combine like terms. x + x + 2 is same as 2x + 2 Pawel Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113? mariel Reply Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113 Pawel how do I set up the problem? Harshika Reply what is a solution set? Harshika find the subring of gaussian integers? Rofiqul hello, I am happy to help! Shirley Reply please can go further on polynomials quadratic Abdullahi hi mam Mark I need quadratic equation link to Alpa Beta Abdullahi Reply find the value of 2x=32 Felix Reply divide by 2 on each side of the equal sign to solve for x corri X=16 Michael Want to review on complex number 1.What are complex number 2.How to solve complex number problems. Beyan yes i wantt to review Mark use the y -intercept and slope to sketch the graph of the equation y=6x Only Reply how do we prove the quadratic formular Seidu Reply please help me prove quadratic formula Darius hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher Shirley Reply thank you help me with how to prove the quadratic equation Seidu may God blessed u for that. Please I want u to help me in sets. Opoku what is math number Tric Reply 4 Trista x-2y+3z=-3 2x-y+z=7 -x+3y-z=6 Sidiki Reply can you teacch how to solve that🙏 Mark Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411 Brenna (61/11,41/11,−4/11) Brenna x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11 Brenna Need help solving this problem (2/7)^-2 Simone Reply x+2y-z=7 Sidiki what is the coefficient of -4× Mehri Reply -1 Shedrak the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1 Alfred Reply A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place. Kimberly Reply Jeannette has$5 and \$10 bills in her wallet. 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