With a perfectly balanced roulette wheel, in the long run,...

With a perfectly balanced roulette wheel, in the long run, red numbers should turn up 18 times in $38 .$ To test its wheel, one casino records the results of $3,800$ plays, finding $1,890$ red numbers. Is that too many reds? Or chance variation? (a) Formulate the null and alternative hypotheses as statements about a box model. (b) The null says that the percentage of reds in the box is alternative says that the percentage of reds in the box is the blanks. (c) Compute $z$ and $P$ . (d) Were there too many reds?

With a perfectly balanced roulette wheel, in the long run, red numbers should
turn up 18 times in $38 .$ To test its wheel, one casino records the results of $3,800$
plays, finding $1,890$ red numbers. Is that too many reds? Or chance variation?
(a) Formulate the null and alternative hypotheses as statements about a box
model.
(b) The null says that the percentage of reds in the box is
alternative says that the percentage of reds in the box is
the blanks.
(c) Compute $z$ and $P$ .
(d) Were there too many reds?

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about a population parameter based on sample data. It involves formulating two competing statements—the null hypothesis and the alternative hypothesis—and then using sample data to decide which hypothesis is more consistent with the observed evidence.

Null Hypothesis

The null hypothesis is the default assumption or claim that there is no effect or difference, and it is presumed to be true until evidence suggests otherwise. In the context of a proportion or fairness test, it often states that the observed proportion is equal to the expected value under a well-behaved or balanced system.

Alternative Hypothesis

The alternative hypothesis is the statement that contradicts the null hypothesis and represents the outcome an analyst wants to support. It posits that there is a significant difference or effect present, such as the proportion in the population being greater than or less than a specified value.

Box Model

The box model is a conceptual tool used in probability to visualize and simulate random processes, where all possible outcomes are represented as members of a box. In hypothesis testing, outcomes (such as red or not red) are thought of as draws from a box with a known distribution under the null hypothesis.

z-Test

The z-test is a statistical test used to determine if there is a significant difference between an observed sample statistic and a known population parameter, under the assumption that the sampling distribution is approximately normal. It involves calculating a z-score, which measures how many standard deviations the observed statistic is away from the hypothesized parameter.

p-Value

The p-value is a probability measure that helps determine the significance of the test results. It quantifies the chance of observing data as extreme as or more extreme than the current sample, assuming that the null hypothesis is true. A small p-value indicates that the observed result is unlikely under the null hypothesis, providing evidence to support the alternative hypothesis.

Transcript

00:01 So we're talking about a roulette wheel here.

00:09 Okay, so as for the question, they have said that it has 38 spaces.

00:16 Okay, now from the 38, we can say that there are 18 red.

00:20 Okay, then there are 18 black and the remaining two are green.

00:29 All right.

00:29 Now, for 3 ,800 spins of the wheel, they have said that the result is 1 ,000.

00:48 890 red numbers.

00:52 Okay.

00:53 So if i now start with the first part, i'm going to start with finding the population proportion here.

01:00 So the population proportion is, i can say it is p and that is given by 18 upon the total is 38.

01:15 So this gives me 47 .4.

01:18 Now the box model i'm going to construct here.

01:20 So the box model is going to be like this so we have a zero here and this is a one here so for the one we've just calculated population proportion as 47 .4 percent so the remaining over here will be 52 .6 percent okay and this we have drawn is for a total of 3 ,800 draws so i'm going to now start considering the sum with the null hypothesis so for the null hypothesis, we say that 47 .4 % are the, i can say 47 % of the spins are red.

02:22 This is 47 .4%.

02:24 And then for the alternate hypothesis, it's going to be opposite of the null.

02:34 So we say that the spins are more than 47 .4%...

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