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College' students and STDs: A recent report estimated that 25% of all college students in the United States have a sexually transmitted disease (STD) Due to th...

Question

College' students and STDs: A recent report estimated that 25% of all college students in the United States have a sexually transmitted disease (STD) Due to the demographics of the community, the director of the campus health center believes that the proportion of students who have a STD is lower at his college: He tests Ho: p = 0.25 versus Ha: p < 0.25.The campus health center staff select a random sample of 50 students and determine that 18% have been with a STD diagnosed Are condition

College' students and STDs: A recent report estimated that 25% of all college students in the United States have a sexually transmitted disease (STD) Due to the demographics of the community, the director of the campus health center believes that the proportion of students who have a STD is lower at his college: He tests Ho: p = 0.25 versus Ha: p < 0.25.The campus health center staff select a random sample of 50 students and determine that 18% have been with a STD diagnosed Are conditions met for use of the normal model to represent the distribution of sample proportions?



Answers

Testing Claims Using P-Values $(a)$ identify the claim and state $H_{0}$ and $H_{a}$, (b) use technology to find the P-value, (c) decide whether to reject or fail to reject the null hypothesis, and (d) interpret the decision in the context of the original claim. Assume the population is normally distributed. Class Size You receive a brochure from a large university. The brochure indicates that the mean class size for full-time faculty is fewer than 32 students. You want to test this claim. You randomly select 18 classes taught by full-time faculty and determine the class size of each. The results are shown in the table at the left. At $\alpha=0.05,$ can you support the university's claim?

Mhm. The proportion of adults that are college grads in Tempe Arizona is estimated at p equals 0.4 is asked this and equals 15 are selected. And if the number is between four and eight hypothesis accepted Otherwise P does not equal 80.4. This question is the language hypothesis testing or test our understanding of this topic. We want to answer a three B accordingly and we wish to find the type one error probability of P equals 10.4. The type one error probability is alpha. That is our significance level. So the critical values are for X equals six plus or minus two or 4/15 at 8 15. Thus alpha level is probably the X is less than 4/15 and greater than 1/15 from a normal distribution. We have teensy one equals P hotlines. Peanut over route peanut online. It's peanut over end. This comes out as negative 1.5. Mg two is 1.5. That's from a standard normal distribution. Alpha spc lessons. The one equals plus pc is greater than D two equals two. Pz lessons. You won by symmetry equals 20.2938. And then we want to find the probability of committing a type two error. The being value with the true proportions 20.2. So this is the probability that we accept asian on given P equals 0.2. That's right. This is the probability that Z is between for 15 minus 150.2 over route 0.2 point 8 15 and 8 15.2 over route 0.2 point eight december 15. This is probably the disease between 150.64 and 3.22 or 0.2604.

Mhm. The proportion of adults that are college grads in Tempe Arizona is estimated at p equals 0.4 is asked this and equals 15 are selected. And if the number is between four and eight hypothesis accepted Otherwise P does not equal 80.4. This question is the language hypothesis testing or test our understanding of this topic. We want to answer a three B accordingly and we wish to find the type one error probability of P equals 10.4. The type one error probability is alpha. That is our significance level. So the critical values are for X equals six plus or minus two or 4/15 at 8 15. Thus alpha level is probably the X is less than 4/15 and greater than 1/15 from a normal distribution. We have teensy one equals P hotlines. Peanut over route peanut online. It's peanut over end. This comes out as negative 1.5. Mg two is 1.5. That's from a standard normal distribution. Alpha spc lessons. The one equals plus pc is greater than D two equals two. Pz lessons. You won by symmetry equals 20.2938. And then we want to find the probability of committing a type two error. The being value with the true proportions 20.2. So this is the probability that we accept asian on given P equals 0.2. That's right. This is the probability that Z is between for 15 minus 150.2 over route 0.2 point 8 15 and 8 15.2 over route 0.2 point eight december 15. This is probably the disease between 150.64 and 3.22 or 0.2604.

The following is a solution for number 12 and this one looks at proportion of private versus public school teachers that have a NBC license. I can't remember what NBC stands for National Bureau of something or other, but license and we want to be 99% confident. We want to have a margin of error of no more than 5% points. So we have two parts here, one where we actually do know what the proportions are. It's about 10.15 or about 15% of the teachers that have licenses in both cases. So let's do that first. We're going to use the proportion formula. So that takes the critical values the star, which we look at the confidence level for that 99% confidence means that that Z Stars 2.576. Now you can get that from a table or from your calculator, but I would really commit that to memory. So 2.576 squared Z star squared and then times p one times one minus P one, P one is 10.15 and then one minus that is 10.85 and then plus P two, which is also 20.15 And then times 1 -22, which again would be .85. Okay. And then we divide that by the margin of error square two if you can't tell. So in equals. So whenever you plug this in, the calculator should get 6 76.8. But whenever you round, you always always always, always have to round whenever you have a sample size and always round up it should be uh 6 77. So 677 is the minimum sample size necessary to achieve this goal. If you know what p. one and p two r. And those are points 15. Now, what do you do whenever it's unknown? Well P one P two. If you don't know it then you have to use 20.5, that's just the general world that maximizes your in. And you'll see it does get pretty big so 2.576 squared. Everything else can stay the same. But instead of using those P one P two values, we have to use 20.5 since we don't know what it is. So P one is .51- people on his .5 Plus P two which is five times 1 -22 which is also five. And we divide that By the .5 squared. So you plug all that stuff in and you get 13 27.2 which again round up, never round off. It's not 13 27 that's not big enough, it's 13 28. So you can see it's quite a bit bigger, the less information, you know, you know? But in fact it's almost twice as large but that is the minimum sample size necessary if you don't know what P one and P two R. And you want to satisfy these can do

Problem. 24 Question eight. It's not that new is smaller than equal toe 3500 and each one equals that mu bigger than 3500. So the the the venue is equal to x bar minus new note over s over square root end, which is 3375135 year review over standard division over square root off end, she is equal to negative 1.571 for question be so. The Big Valley is a number or interpret in the column for Table five with degree of freedom equal to N minus one, which is eight minus one is equal to seven. So accordingly the table that we value is between 4.45 and 4.1 S O. C, which is so the P value is smaller than the significance level than the null hypothesis is rejected. He is bigger than open toe five. So we fail to reject the mile high processes so we can say that there is no sufficient evidence to support the clean


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