Conditions for a goodness of fits test

A chain of stores knows that 90% of its customers pay with a credit or debit card, 5% pay with cash, and 5% pay with a service on their smartphones. They installed a new system for receiving payments, and they wonder if these percentages still hold true. They plan to take a random sample of customers in order to perform a \chi^2χ2\chi, squared goodness-of-fit test on the results. What is the smallest sample size the company can take to pass the large counts condition?

100

Mabel runs a website, and she wonders how people navigate to her website. She suspects that 50% of visitors arrive from a web search, 25% arrive from links on social media, and 25% arrive directly by entering the website's address. She plans to take a random sample of visitors and record how they navigated to the site in order to perform a \chi^2χ2\chi, squared goodness-of-fit test on the results. What is the smallest sample size Mabel can take to pass the large counts condition?

20

Patrick is a health researcher. He wonders if emergency room visits are evenly distributed across the days of the week. He plans to take a random sample of recent visits in order to carry out a \chi^2χ2\chi, squared goodness-of-fit test on the results. What is the smallest sample size Patrick can take to pass the large counts condition?

35

Miriam wants to test if her 10 -sided die is fair. In other words, she wants to test if some sides get rolled more often than others. She plans on recording how often each side appears in a series of rolls and carrying out a \chi^2χ2\chi, squared goodness-of-fit test on the results. What is the smallest sample size Miriam can take to pass the large counts condition?

50

Peter bought a big pack of 360 party balloons. The balloons come in 6 different colors which are supposed to be distributed evenly in the pack. Peter wants to test whether the distribution is indeed even, but he doesn't want to go over the entire pack. So, he plans to take a sample and carry out a \chi^2χ2\chi, squared goodness-of-fit test on the resulting data. Which of these are conditions for carrying out this test?

He takes a random sample of balloons. He expects each color to appear at least 5 times. He samples 36 balloons at most.

Maura's teacher gives multiple choice tests where each question has 4 choices: A, B, C, and D. Maura wonders if some of these choices appear as correct answers more often than others. She plans on taking a sample of her teacher's test questions and tallying how many times each option appears as a correct answer. She wants to carry out a \chi^2χ2\chi, squared goodness-of-fit test on the resulting data. Which of these are conditions for carrying out this test? Choose 2 answers:

She takes a random sample of her teacher's test questions. She expects each option to be used as a correct answer at least 5 times.

Whitney's town has 10,000 residents and three neighborhoods. These are the percentages of each neighborhood's area relative to the town's total area: Whitney wants to test if the distribution of the neighborhoods' populations matches the distribution of the neighborhoods' areas. She plans to ask a sample of residents what neighborhood they live in. She'll carry out a \chi^2χ2\chi, squared goodness-of-fit test on the resulting data. Which of these are conditions for carrying out this test?

She takes a random sample of residents. She expects each neighborhood to appear at least 555 times. She samples 1000 residents at most.

Pauline sits near the snacks in the office. There are 5 flavors of chocolate, and she wonders if some flavors get chosen more than others. She plans to record how often each flavor gets chosen in a sample of selections to carry out a \chi^2χ2\chi, squared goodness-of-fit test on the resulting data. Which of these are conditions for carrying out this test?

She takes a random sample of selections. She expects each snack to be selected at least 555 times.

Andre read a report saying that 35% of people in his country approved of the job their current Prime Minister was doing, 35% disapproved, and 30% neither approved or disapproved. He wondered if these percentages held true in his city, so he obtained a random sample of 16 responses from people in his city. Here are the results: Andre wanted to use these results to carry out a \chi^2χ2\chi, squared goodness-of-fit test to determine if the sample disagreed with the reported distribution. Which count(s) make this sample fail the large counts condition for this test?

The expected count of people who neither approve nor disapprove of the Prime Minister's job.

A company plans on offering a new smartphone in four colors: black, white, silver, and gold. They suspect that 55% of customers prefer black, 20% prefer white, 10% prefer silver, and 15% prefer gold. They take a random sample of 33 potential customers to see what color they prefer. Here are the results: The company wants to use these results to carry out a \chi^2χ2\chi, squared goodness-of-fit test to determine if the sample disagrees with the expected distribution. Which count(s) make this sample fail the large counts condition for this test?

The expected count of people who prefer silver. The expected count of people who prefer gold.

June is a researcher. She read a 2016 study that published the following population distribution for Americans: She wonders if these figures still hold true, so she takes a sample of 38 Americans and records their ages. Here are the results: June wants to use these results to carry out a \chi^2χ2\chi, squared goodness-of-fit test to determine if her sample disagrees with the official percentages. Which count(s) make this sample fail the large counts condition for this test?

The expected count of the 19-25 age group. The expected count of the 26-34 age group. The expected count of the 55-64 age group.

Terrell's company sells candy in packs that are supposed to contain 50% red candies, 25% orange, and 25% yellow. He randomly selected a pack containing 16 candies and counted how many of each color were in the pack. Here are his results: He wants to use these results to carry out a \chi^2χ2\chi, squared goodness-of-fit test to determine if the color distribution disagrees with the target percentages. Which count(s) make this sample fail the large counts condition for this test?

The expected count of yellow candies. The expected count of orange candies.