Researchers studied the behavior of birds that were searching for seeds and insects in an Oregon forest. In this forest, 54% of the trees were Douglas firs, 40% were ponderosa pines, and 6% were other types of trees. At a randomly selected time during the day, the researchers observed 156 red-breasted nuthatches: 70 were seen in Douglas firs, 79 in ponderosa pines, and 7 in other types of trees. Do these data provide convincing evidence that nuthatches prefer particular types of trees when they're searching for seeds and insects?

A company claims that each batch of its deluxe mixed nuts contains 52% cashews, 27% almonds, 13% macadamia nuts, and 8% brazil nuts. To test this claim, a quality-control inspector takes a random sample of 150 nuts from the latest batch. State appropriate hypotheses for performing a test of the company's claim.

Select the best answer. Which of the following is false? (a) A chi-square distribution with k degrees of freedom is more right-skewed than a chi-square distribution with k + 1 degrees of freedom. (b) A chi-square distribution never takes negative values. (c) The degrees of freedom for a chi-square test is determined by the sample size. (d)

$$P(\chi^2 > 10)$$

is greater when df = k + 1 than when df = k. (e) The area under a chi-square density curve is always equal to 1.

A school's principal wants to know if students spend about the same amount of time on homework each night of the week. She asks a random sample of 50 students to keep track of their homework time for a week. The following table displays the average amount of time (in minutes) students reported per night. Explain carefully why it would not be appropriate to perform a chi-square test for goodness of fit using these data.

Kellogg's Froot Loops cereal comes in six colors: orange, yellow, purple, red, blue, and green. Charise randomly selected 120 loops and noted the color of each. Here are her data:

$$\begin{array}{lcccccc} \hline \text { Color } & \text { Orange } & \text { Yellow } & \text { Purple } & \text { Red } & \text { Blue } & \text { Green } \\ \text { Count } & 28 & 21 & 16 & 25 & 14 & 16 \\ \hline \end{array}$$

Do these data provide convincing evidence at the 5% significance level that Kellogg's Froot Loops do not contain an equal proportion of each color?

Researchers want to know if birds prefer particular types of trees when they're searching for seeds and insects. In an Oregon forest, 54% of the trees were Douglas firs, 40% were ponderosa pines, and 6% were other types of trees. At a randomly selected time during the day, the researchers observed 156 red-breasted nuthatches: 70 were seen in Douglas firs, 79 in ponderosa pines, and 7 in other types of trees. Calculate the value of the chi-square test statistic for these data.

Researchers studied the behavior of birds that were searching for seeds and insects in an Oregon forest. In this forest, $54\%$ of the trees were Douglas firs, $40\%$ were ponderosa pines, and $6\%$ were other types of trees. The researchers made 156 observations of red-breasted nuthatches: 70 were seen in Douglas firs, 79 in ponderosa pines, and 7 in other types of trees. Do these data suggest that nuthatches prefer particular types of trees when they're searching for seeds and insects? Carry out a chi-square goodness of fit test to help answer this question.

A company claims that each batch of its deluxe mixed nuts contains 52% cashews, 27% almonds, 13% macadamia nuts, and 8% Brazil nuts. To test this claim, a quality-control inspector takes a random sample of 150 nuts in the latest batch. The table displays the sample data:

$$\begin{array}{lcccc} \hline \text { Type of nut } & \text { Cashew } & \text { Almond } & \text { Macadamia } & \text { Brazil } \\ \text { Count } & 83 & 29 & 20 & 18 \\ \hline \end{array}$$

Calculate the expected count for each type of nut.

Which of the following statements about chi-square distributions are true? I. For all chi-square distributions, $P\left(\chi^{2} \geq 0\right)=1.$ II. A chi-square distribution with fewer than 10 degrees of freedom is roughly symmetric. III. The more degrees of freedom a chi-square distribution has, the larger the mean of the distribution. (a) I only (b) II only (c) III only (d) I and III (e) I, II, and III

Kellogg's Froot Loops cereal comes in 6 colors: orange, yellow, purple, red, blue, and green. Are these colors uniformly distributed? Charise poured out her morning bowl of cereal and methodically counted the number of cereal pieces of each color. State appropriate hypotheses for testing if the proportion of each color of Froot Loops is the same.

The plants in an area have either very small or very large seeds. Birds with small beaks can eat small seeds, and birds with large beaks can eat large seeds. Birds with smaller beaks and birds with larger beaks become more common than birds with medium beaks. What type of selection is this? A. stabilizing selection B. disruptive selection C. directional selection

Researchers want to know if birds prefer particular types of trees when they're searching for seeds and insects. In an Oregon forest, 54% of the trees were Douglas firs, 40% were ponderosa pines, and 6% were other types of trees. At a randomly selected time during the day, the researchers observed 156 red-breasted nuthatches: 70 were seen in Douglas firs, 79 in ponderosa pines, and 7 in other types of trees. State the hypotheses that the researchers are interested in testing.

Researchers want to know if birds prefer particular types of trees when they're searching for seeds and insects. In an Oregon forest, 54 % of the trees are Douglas firs, 40 % are ponderosa pines, and 6 % are other types of trees. At a randomly selected time during the day, the researchers observed 156 red-breasted nuthatches: 70 were seen in Douglas firs, 79 in ponderosa pines, and 7 in other types of trees.

Calculate the expected counts for a test of the null hypothesis in the previous part.

Refer to the earlier exercise randomly selecting the 100 students, the principal personally interviewed the students about their homework habits, including the amount of time they spend doing homework in a typical week.

Why did the principal include a margin of error with her estimate?