According to researchers, a coin flip may not have a 50% chance of landing heads and a 50% chance of landing tails. In fact, they believe that a coin is more likely to land the same way it started. So if it starts out heads up, it is more likely to land heads up. Suppose someone tests this hypoth- esis with 1,000 flips of a coin where it starts out heads up each time. a. Describe what the symbol a stands for in this context. b. State your null and alternative hypotheses. c. Suppose 52% of the sample of 1,000 flips landed heads facing up. Verify the validity conditions that allow us to use a theory-based test. d. A theory-based test reports a standardized statistic of 1.26. Interpret what this means. e. A theory-based test reports a p-value of 0.1030. State your conclusion in terms of strength of evidence and what that means in the context of the study.

a) π = np

π represents the number of heads that turn up in 1000 tosses of the coin.

b) The null hypothesis is represented as

H₀: p ≤ 0.50

The alternative hypothesis is given as

Hₐ: p > 0.50

c) The validity conditions that must be met to be able to perform a theory-based test to test the hypothesis is having a sample size of 20 in each group and the distribution should not be strongly skewed.

The validity conditions are met because we have 1000 tosses with 520 heads and 480 tails, indicating that we have more than 20 sample size in this sample.

The sample proportion (0.52) and the standard error of the sample proportion (0.0158) show that the distribution approximates a normal distribution and isn't skewed. So, the theory based test for this study is valid.

d) Check Explanation

e) The p-value obtained is greater than the significance level at which the test might have been performed, hence, we fail to reject the null hypothesis and conclude that there is no significant evidence that the coin is likely to turn up heads more times when tossed multiple times, starting with a first toss that gives a head.

The researchers' claim then has to be wrong.

Step-by-step explanation:

a) If p corresponds to the proportion of 1000 tosses that turn up heads,

π = np

where n = number of tosses.

b) For hypothesis testing, the first thing to define is the null and alternative hypothesis.

The null hypothesis plays the devil's advocate and usually takes the form of the opposite of the theory to be tested. It usually contains the signs =, ≤ and ≥ depending on the directions of the test.

While, the alternative hypothesis usually confirms the the theory being tested by the experimental setup. It usually contains the signs ≠, < and > depending on the directions of the test.

For this question where we want to verify that the coin is likely to turn up heads more times when tossed multiple times, starting with a first toss that gives a head. That is, the proportion of heads in multiple tosses is more than 0.5 given that the first toss was a head.

The null hypothesis would be that there is no significant evidence that the coin is likely to turn up heads more times when tossed multiple times, starting with a first toss that gives a head.

That is, the coin is likely to turn up heads less than or equal to 50% of the time, when it is tossed multiple times, starting with a first toss that gives a head.

The alternative hypothesis is that there is significant evidence to conclude that the coin is likely to turn up heads more times when tossed multiple times, starting with a first toss that gives a head.

Mathematically,

The null hypothesis is given as

The null hypothesis is represented as

H₀: p ≤ 0.50

The alternative hypothesis is given as

Hₐ: p > 0.50

c) The conditions that need to be satisfied before a theory based test is used include:

The validity conditions that must be met to be able to perform a theory-based test to test the hypothesis is having a sample size of 20 in each group and the distribution should not be strongly skewed.

d) The standardized statistic shows how far away from the standard proportion (the proportion that the population proportion is being compared with) the sample proportion is in terms of the standard error of the sample proportion.

It is given mathematically as,

t or z = (x - μ)/σₓ

x = p = sample proportion of the number of heads obtained in the multiple tosses starting with a first result of a head turning up = 0.52

μ = p₀ = 0.50 (the standard being tested against.

σₓ = standard error of the sample proportion, given as σₓ = √[p(1-p)/n]

n = sample size = 100

σₓ = 0.0158

The standardized statistic is also used to obtain the p-value that indicates how significant the results of the theory based test is.

e) The interpretation of p-values is that

When the (p-value > significance level), we fail to reject the null hypothesis and when the (p-value < significance level), we reject the null hypothesis and accept the alternative hypothesis.

So, for this question, like all other hypothesis testing, the significance level is usually at 0.05. On rare occasions, 0.01 and 0.10 are often used.

Whichever of the 3 is used,

p-value = 0.1030

0.1030 > 0.05, 0.01 or 0.10

Hence,

p-value > significance level

This means that we fail to reject the null hypothesis & conclude that there is no significant evidence that the coin is likely to turn up heads more times when tossed multiple times, starting with a first toss that gives a head.

Hope this Helps!!!