91Ó°ÊÓ

a. In \(1965,\) a newspaper carried a story about a high school student who reported getting 9207 heads and 8743 tails in 17,950 coin tosses. Is this a significant discrepancy from the null hypothesis \(H_{0}: p=\frac{1}{2} ?\) b. Jack Youden, a statistician at the National Bureau of Standards, contacted the student and asked him exactly how he had performed the experiment (Youden 1974). To save time, the student had tossed groups of five coins at a time, and a younger brother had recorded the results, shown in the following table: $$\begin{array}{cc} \hline \text { Number of Heads } & \text { Frequency } \\\ \hline 0 & 100 \\ 1 & 524 \\ 2 & 1080 \\ 3 & 1126 \\ 4 & 655 \\ 5 & 105 \\\ \hline \end{array}$$ Are the data consistent with the hypothesis that all the coins were fair \(\left(p=\frac{1}{2}\right) ?\) c. Are the data consistent with the hypothesis that all five coins had the same probability of heads but that this probability was not necessarily \(\frac{1}{2} ?\) (Hint: Use the binomial distribution.)

Step by step solution

01

Define the Null Hypothesis

For part (a), the null hypothesis is that each coin toss is fair, meaning each has a \(50\%\) probability of being heads or tails: \(H_0: p = \frac{1}{2}\).

02

Calculate Expected Results for Fair Coin

In 17,950 tosses, if the coin is fair, we expect roughly half to be heads, and half to be tails: \(\text{Expected Heads} = 0.5 \times 17950 = 8975\), \(\text{Expected Tails} = 8975\).

03

Analyze Observed vs Expected for Significance

We need to determine whether the observed heads \(9207\) significantly exceeds the expected \(8975\) using statistical tests (e.g., Chi-Square test). Calculate \(\chi^2\) as follows: \[\chi^2 = \frac{(9207-8975)^2}{8975} + \frac{(8743-8975)^2}{8975}\]. If \(\chi^2\) exceeds the critical value at 1 degree of freedom, the discrepancy is significant.

04

Calculate Binomial Probabilities for Part (b)

To analyze part (b), calculate the binomial probabilities for 0 to 5 heads in 5 coin tosses, assuming a fair coin \(p = 0.5\). Use \(P(X = k) = \binom{5}{k} \cdot p^k \cdot (1-p)^{5-k}\) to determine probabilities for each outcome.

05

Expected Frequencies and Chi-Square Test for Part (b)

Multiply the probabilities by the total number of trials (3590 tosses recorded) to find the expected frequency for each number of heads (0 through 5). Calculate \(\chi^2\) to test if the discrepancy between observed and expected frequencies is significant.

06

Testing for Consistent Coins (Part c)

For part (c), assume each coin could be biased but must have the same probability \(p\). Check goodness-of-fit to calculated similar binomial probabilities \(P(X=k)\) using the best-fit \(p\), and apply a Chi-Square test to see how closely the observed data aligns with these probabilities.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Chi-Square Test

A Chi-Square Test is a statistical method used to examine the differences between observed and expected frequencies in a data set. It is particularly useful in determining how likely it is that any observed variance is due to randomness.

In our example, a Chi-Square Test was used to determine if the observed number of heads from coin tosses significantly deviated from what would be expected if all coins were fair. The expected number of heads and tails, based on fair coins, should be equal.

By calculating the \[ \chi^2 = \frac{(\text{Observed Heads} - \text{Expected Heads})^2}{\text{Expected Heads}} + \frac{(\text{Observed Tails} - \text{Expected Tails})^2}{\text{Expected Tails}} \] we can test the null hypothesis that any deviation is due to chance. A significant Chi-Square value indicates that the coins are likely not fair.

Binomial Distribution

A Binomial Distribution is a probability distribution used to model the number of successes in a fixed number of independent trials, where each trial has two possible outcomes (success or failure). It is often used to calculate probabilities when dealing with situations like coin tosses.

If each coin toss is independent and fair, with each having a probability \( p = \frac{1}{2} \) of landing heads, the binomial distribution can help predict the number of heads observed.

The formula \( P(X = k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k} \) is used to calculate the probability of exactly \( k \) heads in \( n \) tosses. In our problem, this distribution helps us check if the outcomes for grouped coin tosses deviate from expected results by comparing the observed and predicted frequencies of heads.

Goodness-of-Fit

The Goodness-of-Fit test is used to see how well observed data fits with the expected data from a specified distribution, such as the binomial distribution. It helps to assess if the model describes the observed data accurately enough.

In the case of coin tossing, where initial assumptions suggest that tosses are fair, the Goodness-of-Fit test helps verify whether the observed distribution of heads per group of tosses aligns with what we'd expect from fair coins.

You calculate expected values using a distribution assumption (like fair coins) and then check with a Chi-Square Test to see if the discrepancy between actual and expected counts can be attributed to random chance or if they are significant enough to suggest the assumption was wrong. This process helps verify if a specific parameter, like the probability of heads, fits the observed data well.