# The homebuilder’s association reports that 75% of home buyers would like a fireplace in their new home. A local builder finds that 20 out of 30 of his customers wanted a fireplace. Using the p value decision rule determine at the 5% level of significance is there is enough evidence from the sample to conclude that the homebuilders report is too high? If P is too high in displaystyle{H}_{{0}} then what should it be in displaystyle{H}_{{1}}? Also construct a displaystyle{left({1}-alpharight)}% interval estimate for the true proportion P. Make sure to state the meaning of P first…and completely interpret the CI for P.

Question
Confidence intervals
The homebuilder’s association reports that $$75\%$$ of home buyers would like a fireplace in their new home. A local builder finds that 20 out of 30 of his customers wanted a fireplace. Using the p value decision rule determine at the 5% level of significance is there is enough evidence from the sample to conclude that the homebuilders report is too high?
If P is too high in $$\displaystyle{H}_{{0}}$$ then what should it be in $$\displaystyle{H}_{{1}}$$?
Also construct a $$\displaystyle{\left({1}-\alpha\right)}\%$$ interval estimate for the true proportion P.
Make sure to state the meaning of P first…and completely interpret the CI for P.

2021-03-09
Step 1
According to the homebuilder's report, the population proportion is $$75\%$$, but on the basis of the sample selected, the sample proportion (an unbiased estimate of population proportion) is $$\displaystyle{20}\text{/}{30}={66.67}\%$$
Therefore, the hypothesis are given as:
$$\displaystyle{H}_{{0}}:{P}={0.75}$$
$$\displaystyle{H}_{{1}}:{P}<{0.75}$$</span>
To test the above claim, standard normal distribution (z-test) will be used. The formula for the same is given by:
$$\displaystyle{z}=\frac{{{p}-{P}}}{\sqrt{{\frac{{{P}{Q}}}{{n}}}}}\approx{N}{\left({0},{1}\right)}$$
$$\displaystyle{z}=\frac{{{0.6667}-{0.75}}}{\sqrt{{\frac{{{\left({0.75}\right)}{\left({0.25}\right)}}}{{30}}}}}$$
$$\displaystyle=\frac{{-{0.0833}}}{{0.0791}}$$
$$\displaystyle=-{1.054}$$
The critical z - value at $$\displaystyle\alpha={0.05}$$ (left - tailed) is -1.645
The p - value which is the probability value for rejecting null hypothesis is 0.1469
According to the p-value approach, null hypothesis is rejected when p-value is less than the $$\displaystyle\alpha$$ level of significance and is accepted otherwise.
$$\displaystyle{0.1469}>{0.05}-\text{Accept}-{H}_{{0}}$$
Therefore, there are insufficient evidence to reject null hypothesis, concluding that the homebuilders report is not too high and that the population proportion for home buyers who would like fireplace in their house is $$0.75 (75\%)$$.
Step 2
The formula for $$\displaystyle{\left({1}-\alpha\right)}\%$$ confidence interval for population proportion (P) is given by:
$$\displaystyle{C}.{I}={p}\pm{z}_{{\alpha\text{/}{2}}}.{S}{E}{\left({p}\right)}$$
$$\displaystyle{p}=$$ sample proportion
$$\displaystyle{z}_{{\alpha\text{/}{2}}}=\text{critical}\ {z}-\text{value at}\ \alpha={0.025}{\left({0.05}\text{/}{2}\right)}$$ level of significance
$$\displaystyle{S}{E}{\left({p}\right)}={s} \tan{}$$ dard error for sample proportion
$$\displaystyle{p}={0.6667}$$
$$\displaystyle{z}_{{\alpha\text{/}{2}}}=\pm{1.96}$$
$$\displaystyle{S}{E}{\left({p}\right)}=\sqrt{{\frac{{{p}{\left({1}-{p}\right)}}}{{n}}}}$$
$$\displaystyle=\sqrt{{\frac{{{\left({0.6667}\right)}{\left({1}-{0.6667}\right)}}}{{30}}}}$$
$$\displaystyle={0.086}$$
$$\displaystyle{C}.{I}={\left[{0.6667}\pm{\left({1.96}\right)}{\left({0.086}\right)}\right]}$$
$$\displaystyle={\left[{0.4981},{0.8353}\right]}$$
This implies that we are $$95\%$$ confident that the true population proportion is contained by this interval.

### Relevant Questions

A new thermostat has been engineered for the frozen food cases in large supermarkets. Both the old and new thermostats hold temperatures at an average of $$25^{\circ}F$$. However, it is hoped that the new thermostat might be more dependable in the sense that it will hold temperatures closer to $$25^{\circ}F$$. One frozen food case was equipped with the new thermostat, and a random sample of 21 temperature readings gave a sample variance of 5.1. Another similar frozen food case was equipped with the old thermostat, and a random sample of 19 temperature readings gave a sample variance of 12.8. Test the claim that the population variance of the old thermostat temperature readings is larger than that for the new thermostat. Use a $$5\%$$ level of significance. How could your test conclusion relate to the question regarding the dependability of the temperature readings? (Let population 1 refer to data from the old thermostat.)
(a) What is the level of significance?
State the null and alternate hypotheses.
$$H0:?_{1}^{2}=?_{2}^{2},H1:?_{1}^{2}>?_{2}^{2}H0:?_{1}^{2}=?_{2}^{2},H1:?_{1}^{2}\neq?_{2}^{2}H0:?_{1}^{2}=?_{2}^{2},H1:?_{1}^{2}?_{2}^{2},H1:?_{1}^{2}=?_{2}^{2}$$
(b) Find the value of the sample F statistic. (Round your answer to two decimal places.)
What are the degrees of freedom?
$$df_{N} = ?$$
$$df_{D} = ?$$
What assumptions are you making about the original distribution?
The populations follow independent normal distributions. We have random samples from each population.The populations follow dependent normal distributions. We have random samples from each population.The populations follow independent normal distributions.The populations follow independent chi-square distributions. We have random samples from each population.
(c) Find or estimate the P-value of the sample test statistic. (Round your answer to four decimal places.)
(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis?
At the ? = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.At the ? = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant. At the ? = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.At the ? = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.
(e) Interpret your conclusion in the context of the application.
Reject the null hypothesis, there is sufficient evidence that the population variance is larger in the old thermostat temperature readings.Fail to reject the null hypothesis, there is sufficient evidence that the population variance is larger in the old thermostat temperature readings. Fail to reject the null hypothesis, there is insufficient evidence that the population variance is larger in the old thermostat temperature readings.Reject the null hypothesis, there is insufficient evidence that the population variance is larger in the old thermostat temperature readings.
The International Business Society conducts a series of managerial surveys to detect challenges that international managers encounter and what strategies help them overcome those challenges. A previous study showed that employee retention is the most critical organizational challenge currently confronted by international managers. One strategy that may impact employee retention is a successful employee recognition program. The International Business Society randomly selected 423 small firms and 192 large firms. The study showed that 326 of the 423 small firms have employee retention programs compared to 167 of the 192 large organizations.
a) At the 0.01 level of significance, is there evidence of a significant difference between small firms and large firms concerning the proportion that has employee recognition program?
b) Find the p-value and interpret its meaning?
c) Construct and interpret a $$99\%$$ confidence interval estimate for the difference between small firms and large firms concerning the proportion of employee recognition program.
(a) The company's production equipment produces metal discs weighing 200 g. It should be noted that the weight of the discs corresponds to the normal distribution. To check machine consistency, 20 discs are randomly selected with an average weight of 205 g and a standard deviation of 7 g. What is the 99% confidence interval for the average weight of the selected discs?
(b) A company launched a new model of golf ball. It claimed that the driving distance is at least 300m. A sample of 20 balls yields a sample mean of 295m and sample standard deviation of 8m. It is assumed that the driving distance is normally distributed.
Conduct an appropriate hypothesis testing at the 0.05 level of significance. Is there any evidence that the average travel distance stated by the company is true?
Case: Dr. Jung’s Diamonds Selection
With Christmas coming, Dr. Jung became interested in buying diamonds for his wife. After perusing the Web, he learned about the “4Cs” of diamonds: cut, color, clarity, and carat. He knew his wife wanted round-cut earrings mounted in white gold settings, so he immediately narrowed his focus to evaluating color, clarity, and carat for that style earring.
After a bit of searching, Dr. Jung located a number of earring sets that he would consider purchasing. But he knew the pricing of diamonds varied considerably. To assist in his decision making, Dr. Jung decided to use regression analysis to develop a model to predict the retail price of different sets of round-cut earrings based on their color, clarity, and carat scores. He assembled the data in the file Diamonds.xls for this purpose. Use this data to answer the following questions for Dr. Jung.
1) Prepare scatter plots showing the relationship between the earring prices (Y) and each of the potential independent variables. What sort of relationship does each plot suggest?
2) Let X1, X2, and X3 represent diamond color, clarity, and carats, respectively. If Dr. Jung wanted to build a linear regression model to estimate earring prices using these variables, which variables would you recommend that he use? Why?
3) Suppose Dr. Jung decides to use clarity (X2) and carats (X3) as independent variables in a regression model to predict earring prices. What is the estimated regression equation? What is the value of the R2 and adjusted-R2 statistics?
4) Use the regression equation identified in the previous question to create estimated prices for each of the earring sets in Dr. Jung’s sample. Which sets of earrings appear to be overpriced and which appear to be bargains? Based on this analysis, which set of earrings would you suggest that Dr. Jung purchase?
5) Dr. Jung now remembers that it sometimes helps to perform a square root transformation on the dependent variable in a regression problem. Modify your spreadsheet to include a new dependent variable that is the square root on the earring prices (use Excel’s SQRT( ) function). If Dr. Jung wanted to build a linear regression model to estimate the square root of earring prices using the same independent variables as before, which variables would you recommend that he use? Why?
1
6) Suppose Dr. Jung decides to use clarity (X2) and carats (X3) as independent variables in a regression model to predict the square root of the earring prices. What is the estimated regression equation? What is the value of the R2 and adjusted-R2 statistics?
7) Use the regression equation identified in the previous question to create estimated prices for each of the earring sets in Dr. Jung’s sample. (Remember, your model estimates the square root of the earring prices. So you must actually square the model’s estimates to convert them to price estimates.) Which sets of earring appears to be overpriced and which appear to be bargains? Based on this analysis, which set of earrings would you suggest that Dr. Jung purchase?
8) Dr. Jung now also remembers that it sometimes helps to include interaction terms in a regression model—where you create a new independent variable as the product of two of the original variables. Modify your spreadsheet to include three new independent variables, X4, X5, and X6, representing interaction terms where: X4 = X1 × X2, X5 = X1 × X3, and X6 = X2 × X3. There are now six potential independent variables. If Dr. Jung wanted to build a linear regression model to estimate the square root of earring prices using the same independent variables as before, which variables would you recommend that he use? Why?
9) Suppose Dr. Jung decides to use color (X1), carats (X3) and the interaction terms X4 (color * clarity) and X5 (color * carats) as independent variables in a regression model to predict the square root of the earring prices. What is the estimated regression equation? What is the value of the R2 and adjusted-R2 statistics?
10) Use the regression equation identified in the previous question to create estimated prices for each of the earring sets in Dr. Jung’s sample. (Remember, your model estimates the square root of the earring prices. So you must square the model’s estimates to convert them to actual price estimates.) Which sets of earrings appear to be overpriced and which appear to be bargains? Based on this analysis, which set of earrings would you suggest that Dr. Jung purchase?
1. A researcher is interested in finding a 98% confidence interval for the mean number of times per day that college students text. The study included 144 students who averaged 44.7 texts per day. The standard deviation was 16.5 texts. a. To compute the confidence interval use a ? z t distribution. b. With 98% confidence the population mean number of texts per day is between and texts. c. If many groups of 144 randomly selected members are studied, then a different confidence interval would be produced from each group. About percent of these confidence intervals will contain the true population number of texts per day and about percent will not contain the true population mean number of texts per day. 2. You want to obtain a sample to estimate how much parents spend on their kids birthday parties. Based on previous study, you believe the population standard deviation is approximately $$\displaystyle\sigma={40.4}$$ dollars. You would like to be 90% confident that your estimate is within 1.5 dollar(s) of average spending on the birthday parties. How many parents do you have to sample? n = 3. You want to obtain a sample to estimate a population mean. Based on previous evidence, you believe the population standard deviation is approximately $$\displaystyle\sigma={57.5}$$. You would like to be 95% confident that your estimate is within 0.1 of the true population mean. How large of a sample size is required?
factor in determining the usefulness of an examination as a measure of demonstrated ability is the amount of spread that occurs in the grades. If the spread or variation of examination scores is very small, it usually means that the examination was either too hard or too easy. However, if the variance of scores is moderately large, then there is a definite difference in scores between "better," "average," and "poorer" students. A group of attorneys in a Midwest state has been given the task of making up this year's bar examination for the state. The examination has 500 total possible points, and from the history of past examinations, it is known that a standard deviation of around 60 points is desirable. Of course, too large or too small a standard deviation is not good. The attorneys want to test their examination to see how good it is. A preliminary version of the examination (with slight modifications to protect the integrity of the real examination) is given to a random sample of 20 newly graduated law students. Their scores give a sample standard deviation of 70 points. Using a 0.01 level of significance, test the claim that the population standard deviation for the new examination is 60 against the claim that the population standard deviation is different from 60.
(a) What is the level of significance?
State the null and alternate hypotheses.
$$H_{0}:\sigma=60,\ H_{1}:\sigma\ <\ 60H_{0}:\sigma\ >\ 60,\ H_{1}:\sigma=60H_{0}:\sigma=60,\ H_{1}:\sigma\ >\ 60H_{0}:\sigma=60,\ H_{1}:\sigma\ \neq\ 60$$
(b) Find the value of the chi-square statistic for the sample. (Round your answer to two decimal places.)
What are the degrees of freedom?
What assumptions are you making about the original distribution?
We assume a binomial population distribution.We assume a exponential population distribution. We assume a normal population distribution.We assume a uniform population distribution.
A manager at ACME Equipment Sale and Rental wondered how offering a free two-year service warranty on its tractors might influence sales. For the next 500 customers who expressed interest in buying a tractor, 250 were randomly offered a warranty, and the rest did not receive. Ninety-three of those offering a warranty, and fifty-four of those not offering a warranty ended up buying a tractor.
a. Construct a $$95\%$$ confidence interval for the difference between the proportions of customers purchasing tractors with and without warranties. Be sure to check all necessary assumptions and interpret the interval.
b. Test the hypothesis that offering the warranty increases the proportion of customers who eventually purchase a tractor. Be sure to check all necessary assumptions, state the null and alternative hypotheses, obtain the p-value, and state your conclusion. Should a manager offer a warranty based on this test?
A poll in 2017 reported that 699 out of 1027 adults in a certain country believe that marijuana should be legalized. When this poll about the subject was first conducted in 1969, only 12% of rhe country supported legalizztion. Assume the conditions for using the CLT are met.
a) Find and interpet a 99% confidence interval for the proportion of adults in the country 2017 that believe marijuana should be legalized is (0.643, 0.718)
b) Find and interpret a 90%confidence interval for this population parameter. The 90% confidence interval for the proportion of adults in the country 2017 that believe marijuana should be legalized is (0.657, 0.705)
c)Find the margin of error for each of the confidence intervals found The margin of error of the 99% confidence interval is 0.039 and the margin of error of the 90% confidence interval is 0.025
d) Without computing it, how would the margin of error of an 80% confidence interval compare with the margin of error for 90% and 99% intervals? Construct the 80% confidence interval to see if your production was correct
How would a 80% interval compare with the others in the margin of error?
Would you rather spend more federal taxes on art? Of a random sample of $$n_{1} = 86$$ politically conservative voters, $$r_{1} = 18$$ responded yes. Another random sample of $$n_{2} = 85$$ politically moderate voters showed that $$r_{2} = 21$$ responded yes. Does this information indicate that the population proportion of conservative voters inclined to spend more federal tax money on funding the arts is less than the proportion of moderate voters so inclined? Use $$\alpha = 0.05.$$ (a) State the null and alternate hypotheses. $$H_0:p_{1} = p_{2}, H_{1}:p_{1} > p_2$$
$$H_0:p_{1} = p_{2}, H_{1}:p_{1} < p_2$$
$$H_0:p_{1} = p_{2}, H_{1}:p_{1} \neq p_2$$
$$H_{0}:p_{1} < p_{2}, H_{1}:p_{1} = p_{2}$$ (b) What sampling distribution will you use? What assumptions are you making? The Student's t. The number of trials is sufficiently large. The standard normal. The number of trials is sufficiently large.The standard normal. We assume the population distributions are approximately normal. The Student's t. We assume the population distributions are approximately normal. (c)What is the value of the sample test statistic? (Test the difference $$p_{1} - p_{2}$$. Do not use rounded values. Round your final answer to two decimal places.) (d) Find (or estimate) the P-value. (Round your answer to four decimal places.) (e) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level alpha? At the $$\alpha = 0.05$$ level, we reject the null hypothesis and conclude the data are statistically significant. At the $$\alpha = 0.05$$ level, we fail to reject the null hypothesis and conclude the data are statistically significant. At the $$\alpha = 0.05$$ level, we fail to reject the null hypothesis and conclude the data are not statistically significant. At the $$\alpha = 0.05$$ level, we reject the null hypothesis and conclude the data are not statistically significant. (f) Interpret your conclusion in the context of the application. Reject the null hypothesis, there is sufficient evidence that the proportion of conservative voters favoring more tax dollars for the arts is less than the proportion of moderate voters. Fail to reject the null hypothesis, there is sufficient evidence that the proportion of conservative voters favoring more tax dollars for the arts is less than the proportion of moderate voters. Fail to reject the null hypothesis, there is insufficient evidence that the proportion of conservative voters favoring more tax dollars for the arts is less than the proportion of moderate voters. Reject the null hypothesis, there is insufficient evidence that the proportion of conservative voters favoring more tax dollars for the arts is less than the proportion of moderate voters.
What is the optimal time for a scuba diver to be on the bottom of the ocean? That depends on the depth of the dive. The U.S. Navy has done a lot of research on this topic. The Navy defines the "optimal time" to be the time at each depth for the best balance between length of work period and decompression time after surfacing. Let $$\displaystyle{x}=$$ depth of dive in meters, and let $$\displaystyle{y}=$$ optimal time in hours. A random sample of divers gave the following data.
$$\begin{array}{|c|c|} \hline x & 13.1 & 23.3 & 31.2 & 38.3 & 51.3 &20.5 & 22.7 \\ \hline y & 2.78 & 2.18 & 1.48 & 1.03 & 0.75 & 2.38 & 2.20 \\ \hline \end{array}$$
(a)
Find $$\displaystyleΣ{x},Σ{y},Σ{x}^{2},Σ{y}^{2},Σ{x}{y},{\quad\text{and}\quad}{r}$$. (Round r to three decimal places.)
$$\displaystyleΣ{x}=$$
$$\displaystyleΣ{y}=$$
$$\displaystyleΣ{x}^{2}=$$
$$\displaystyleΣ{y}^{2}=$$
$$\displaystyleΣ{x}{y}=$$
$$\displaystyle{r}=$$
(b)
Use a $$1\%$$ level of significance to test the claim that $$\displaystyle\rho<{0}$$. (Round your answers to two decimal places.)
$$\displaystyle{t}=$$
critical $$\displaystyle{t}=$$
Conclusion
Reject the null hypothesis. There is sufficient evidence that $$\displaystyle\rho<{0}$$.Reject the null hypothesis. There is insufficient evidence that $$\displaystyle\rho<{0}$$.
Fail to reject the null hypothesis. There is sufficient evidence that $$\displaystyle\rho<{0}$$.Fail to reject the null hypothesis. There is insufficient evidence that $$\displaystyle\rho<{0}.$$
(c)
Find $$\displaystyle{S}_{{e}},{a},{\quad\text{and}\quad}{b}$$. (Round your answers to four decimal places.)
$$\displaystyle{S}_{{e}}=$$
$$\displaystyle{a}=$$
$$\displaystyle{b}=$$
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