# Bivariate data questions and answers

Recent questions in Bivariate numerical data
Bivariate numerical data

### For all these problems, carry at least four digits after the decimal in your calculations. Answers may vary slightly due to rounding. In order to use a normal distribution to compute confidence intervals for p, what conditions on pp and nq need to be satisfied?

Bivariate numerical data

### The pathogen Phytophthora capsici causes bell pepper plants to wilt and die. A research project was designed to study the effect of soil water content and the spread of the disease in fields of bell peppers. It is thought that too much water helps spread the disease. The fields were divided into rows and quadrants. The soil water content (percent of water by volume of soil) was determined for each plot. An important first step in such a research project is to give a statistical description of the data. Soil Water Content for Bell Pepper Study 1591510amp;14amp;15amp;12amp;10amp;14amp;12amp;9amp;10amp;14amp;9amp;10amp;11amp;13amp;10amp;9amp;9amp;12amp;7amp;9amp;amp;11amp;14amp;16amp;amp;11amp;13amp;16amp;amp;11amp;14amp;12amp;amp;11amp;8amp;10amp;amp;10amp;9amp;11amp;amp;11amp;8amp;11amp;amp;13amp;11amp;12amp;amp;16amp;13amp;15amp;amp;10amp;13amp;6 (a) Make a box-and-whisker plot of the data. Find the interquartile range.

Bivariate numerical data

### You were asked about advantages of using box plots and dot plots to describe and compare distributions of scores. Do you think the advantages you found would exist not only for these data, but for numerical data in general? Explain.

Bivariate numerical data

### Construct 90% and 95% confidence intervals for the population proportion. Interpret the results and compare the widths of the confidence intervals. In a survey of 2241 U.S. adults in a recent year, 650 made a New Year's resolution to eat healthier.

Bivariate numerical data

### Use a graphing calculator to construct a 95% confidence interval for a sample of size 30 from a uniform distribution over the interval (0, 1). Take a class poll to determine the percentage of confidence intervals that contain the true mean. Discuss the result in class.

Bivariate numerical data

### The U.S. Census Bureau compiles census data on educational attainment of Americans. From the document Current Population Reports, we obtained the 2000 distribution of educational attainment for U.S. adults 25 years old and older. Here is that distribution. [Table] A random sample of 500 U.S. adults (25 years old and older) taken this year gave the following frequency distribution. [Table] Decide, at the 5% significance level, whether this year’s distribution of educational attainment differs from the 2000 distribution.

Bivariate numerical data

### The following advanced exercise use a generalized ratio test to determine convergence of some series that arise in particular applications, including the ratio and root test, are not powerful enough to determine their convergence. The test states that if $$$\displaystyle\lim{\left\lbrace{n}\rightarrow\infty\right\rbrace}{\frac{{{a}{\left\lbrace{2}{n}\right\rbrace}}}{{{a}_{{{n}}}}}}{<}\frac{{1}}{{2}}$$ then $$\displaystyle\sum{a}_{{{n}}}$$ converges, while if $$\displaystyle\lim{\left\lbrace{n}\rightarrow\infty\right\rbrace}{\frac{{{a}{\left\lbrace{2}{n}+{1}\right\rbrace}}}{{{a}_{{{n}}}}}}{>}\frac{{1}}{{2}}$$ then $$\displaystyle\sum{a}_{{{n}}}$$$ diverges. Let $$\displaystyle{a}_{{{n}}}={\frac{{{n}^{{{\ln{{n}}}}}}}{{{\left({\ln{{n}}}\right)}^{{{n}}}}}}$$. Show that $$\frac{a_{2n}}{a_{n}} \rightarrow 0$$ as $$\displaystyle{n}\rightarrow\infty$$

Bivariate numerical data

### When a correlation value is reported in research journals, there often is not an accompanying scatterplot. Explain why reported correlation values should be supported with either a scatterplot or a description of the scatterplot.

Bivariate numerical data

### Suppose you were to collect data for each pair of variables. You want to make a scatterplot. Which variable would you use as the explanatory variable and which as the response variable? Why? What would you expect to see in the scatterplot? Discuss the likely direction, form, and strength. Gasoline: number of miles you drove since filling up, gallons remaining in your tank

Bivariate numerical data

### The two-way table summarizes data from an experiment comparing the effectiveness of three different diets (A, B, and C) on weight loss. Researchers randomly assigned 300 volunteer subjects to the three diets. The response variable was whether each subject lost weight over a 1-year period. $$\text{Diet}\ \text{Lost weight?}$$ $$\begin{array}{l|c|c|c|c} & \mathrm{A} & \mathrm{B} & \mathrm{C} & \text { Total } \\ \hline \text { Yes } & & 60 & & 180 \\ \hline \text { No } & & 40 & & 120 \\ \hline \text { Total } & 90 & 100 & 110 & 300 \end{array}$$ Suppose we randomly select one of the subjects from the experiment. Show that the events "Diet B" and "Lost weight" are independent.

Bivariate numerical data

### Driven by technological advances and financial pressures, the number of surgeries performed in physicians' offices nationwide has been increasing over the years. The function $$\displaystyle{f{{\left({t}\right)}}}=-{0.00447}{t}^{{{3}}}+{0.09864}{t}^{{{2}}}+{0.05192}{t}+{0.8}{\left({0}\leq{t}\leq{15}\right)}$$ gives the number of surgeries (in millions) performed in physicians' offices in year t, with $$t=0$$ corresponding to the beginning of 1986. a. Plot the graph of f in the viewing window $$[0,15]\times [0,10]$$. b. Prove that f is increasing on the interval [0, 15].

Bivariate numerical data

### One hundred adults and children were randomly selected and asked whether they spoke more than one language fluently. The data were recorded in a two-way table. Maria and Brennan each used the data to make the tables of joint relative frequencies shown below, but their results are slightly different. The difference is shaded. Can you tell by looking at the tables which of them made an error? Explain. Maria's table YesNo Children0.150.25 Adults0.10.6 Brennan’s table YesNo Children0.150.25 Adults0.10.5

Bivariate numerical data

### You are given the sample mean and the population standard deviation. Use this information to construct 90% and 95% confidence intervals for the population mean. Interpret the results and compare the widths of the confidence intervals. From a random sample of 24 months from January 2006 through December 2016, the mean number of tornadoes per month in the United States was about 100. Assume the population standard deviation is 114.

Bivariate numerical data

### Consider a random sample of size n = 31, with sample mean $$\displaystyle\overline{{{x}}}={45.2}$$ and sample standard deviation s = 5.3. Compute 90%, 95%, and 99% confidence intervals for \muμ using Method 1 with a Student's t distribution. Round endpoints to two digits after the decimal.

Bivariate numerical data

### The following advanced exercise use a generalized ratio test to determine convergence of some series that arise in particular applications, including the ratio and root test, are not powerful enough to determine their convergence. The test states that if  $$\displaystyle\lim{\left\lbrace{n}\rightarrow\infty\right\rbrace}{\frac{{{a}{\left\lbrace{2}{n}\right\rbrace}}}{{{a}_{{{n}}}}}}{<}\frac{{1}}{{2}}$$ then $$\sum a_{n}$$converges,while if $$\displaystyle\lim{\left\lbrace{n}\rightarrow\infty\right\rbrace}{\frac{{{a}{\left\lbrace{2}{n}+{1}\right\rbrace}}}{{{a}_{{{n}}}}}}{>}\frac{{1}}{{2}}$$, then $$\sum a_{n}$$ diverges. Let $$\displaystyle{a}_{{{n}}}={\frac{{{1}}}{{{1}+{x}}}}{\frac{{{2}}}{{{2}+{x}}}}\ldots{\frac{{{n}}}{{{n}+{x}}}}{\frac{{{1}}}{{{n}}}}={\frac{{{\left({n}-{1}\right)}!}}{{{\left({1}+{x}\right)}{\left({2}+{x}\right)}\ldots{\left({n}+{x}\right)}}}}$$. Show that $$\frac{a_{2 n}}{a_{n}} \leq \frac{e^{-x / 2}}{2}$$ . For which x > 0 does the generalized ratio test imply convergence of $$\sum_{n=1}^\infty a_{n}$$?

Bivariate numerical data

### B. G. Cosmos, a scientist, believes that the probability is $$\displaystyle{\frac{{{2}}}{{{5}}}}$$ that aliens from an advanced civilization on Planet X are trying to communicate with us by sending high-frequency signals to Earth. By using sophisticated equipment, Cosmos hopes to pick up these signals. The manufacturer of the equipment, Trekee, Inc., claims that if aliens are indeed sending signals, the probability that the equipment will detect them is $$\displaystyle{\frac{{{3}}}{{{5}}}}$$. However, if aliens are not sending signals, the probability that the equipment will seem to detect such signals is $$\frac{1}{10}$$. If the equipment detects signals, what is the probability that aliens are actually sending them?

Bivariate numerical data

### Consider binary strings with n digits (for example, if n=4, some of the possible strings are 0011,1010,1101, etc.) Let Zn be the number of binary strings of length n that do not contain the substring 000 Find a recurrence relation for Zn You are not required to find a closed form for this recurrence.

Bivariate numerical data

### Find the regression line using the given points.

Bivariate numerical data