professor jennings claims that only 35% of the students at flora college work while attending school. dean…

professor jennings claims that only 35% of the students at flora college work while attending school. dean renata thinks that the professor has underestimated the number of students with part - time or full - time jobs. a random sample of 81 students shows that 37 have jobs. do the data indicate that more than 35% of the students have jobs? (use a 5% level of significance.)\n(a) what are we testing in this problem?\n - a paired difference\n - a single proportion\n - a difference of means\n - a single mean\n - a difference of proportions\n(b) what is the level of significance? indicate the null and alternate hypotheses. (enter != for ≠ as needed.)\n - h0:\n - h1:\n(c) what sampling distribution will you use? what assumptions are you making?\n - well use the students t. well assume that a normal distribution is an appropriate model for the probability distribution of the number of students in the sample with jobs and that the number of students in the sample is not too large so that we can use the normal approximation.\n - well use the standard normal. well assume that a binomial experiment is an appropriate model for the probability distribution of the number of students in the sample with jobs and that the number of students in the sample is not too large so that we can use the normal approximation.\n - well use the standard normal. well assume that a normal distribution is an appropriate model for the probability distribution of the number of students in the sample with jobs and that the number of students in the sample is sufficiently large so that we can use the normal approximation.\n - well use the students t. well assume that a binomial experiment is an appropriate model for the probability distribution of the number of students in the sample with jobs and that the number of students in the sample is sufficiently large so that we can use the normal approximation.\n(d) find the p - value of the test statistic. (round your answer to four decimal places.)\n(e) sketch the sampling distribution and show the area corresponding to the p - value.

professor jennings claims that only 35% of the students at flora college work while attending school. dean renata thinks that the professor has underestimated the number of students with part - time or full - time jobs. a random sample of 81 students shows that 37 have jobs. do the data indicate that more than 35% of the students have jobs? (use a 5% level of significance.)\n(a) what are we testing in this problem?\n - a paired difference\n - a single proportion\n - a difference of means\n - a single mean\n - a difference of proportions\n(b) what is the level of significance? indicate the null and alternate hypotheses. (enter != for ≠ as needed.)\n - h0:\n - h1:\n(c) what sampling distribution will you use? what assumptions are you making?\n - well use the students t. well assume that a normal distribution is an appropriate model for the probability distribution of the number of students in the sample with jobs and that the number of students in the sample is not too large so that we can use the normal approximation.\n - well use the standard normal. well assume that a binomial experiment is an appropriate model for the probability distribution of the number of students in the sample with jobs and that the number of students in the sample is not too large so that we can use the normal approximation.\n - well use the standard normal. well assume that a normal distribution is an appropriate model for the probability distribution of the number of students in the sample with jobs and that the number of students in the sample is sufficiently large so that we can use the normal approximation.\n - well use the students t. well assume that a binomial experiment is an appropriate model for the probability distribution of the number of students in the sample with jobs and that the number of students in the sample is sufficiently large so that we can use the normal approximation.\n(d) find the p - value of the test statistic. (round your answer to four decimal places.)\n(e) sketch the sampling distribution and show the area corresponding to the p - value.

Answer

Explanation:

Step1: Identify the test type

We are testing a single - proportion as we are comparing the proportion of students with jobs in the population (claimed by professor) and in the sample.

Step2: Determine significance level

The level of significance is given as $\alpha = 0.01$.

Step3: State hypotheses

The null hypothesis $H_0:p = 0.32$ (the professor's claim). The alternative hypothesis $H_1:p>0.32$ (Dean Linda's belief that the professor underestimated).

Step4: Select sampling distribution

We will use the standard normal distribution. We assume that a binomial experiment is an appropriate model for the probability distribution of the number of students in the sample with jobs and that the number of students in the sample is sufficiently large so that we can use the normal approximation. The conditions for normal approximation to binomial are $np\geq5$ and $n(1 - p)\geq5$, where $n = 83$ and $p = 0.32$. $np=83\times0.32 = 26.56\geq5$ and $n(1 - p)=83\times(1 - 0.32)=56.44\geq5$.

Step5: Calculate test - statistic

The sample proportion $\hat{p}=\frac{37}{83}\approx0.4458$. The test - statistic $z=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}}=\frac{0.4458 - 0.32}{\sqrt{\frac{0.32\times(1 - 0.32)}{83}}}=\frac{0.1258}{\sqrt{\frac{0.32\times0.68}{83}}}=\frac{0.1258}{\sqrt{\frac{0.2176}{83}}}=\frac{0.1258}{\sqrt{0.0026229}}=\frac{0.1258}{0.0512}\approx2.46$.

Step6: Find p - value

For a right - tailed test with $z = 2.46$, the p - value is $P(Z>2.46)=1 - P(Z\leq2.46)$. From the standard normal table, $P(Z\leq2.46)=0.9931$, so the p - value is $1 - 0.9931 = 0.0069$.

Answer:

(a) A single proportion (b) $\alpha = 0.01$, $H_0:p = 0.32$, $H_1:p>0.32$ (c) Standard normal distribution. Assumption: Binomial experiment is appropriate and sample size is large enough for normal approximation ($np\geq5$ and $n(1 - p)\geq5$) (d) $z\approx2.46$ (e) $p - value\approx0.0069$