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Spinning a coin, unlike tossing it, may not give heads and tails equal probabilities. I spun a penny 150 times and got 67 heads. We wish to find how significant is this evidence against equal probabilities.


AWhat is the sample proportion of heads? (Round to 3 places)


BHeads do not make up half of the sample. Is this sample evidence that the probabilities of heads and tails are different?



Take $p$ to be the probability of getting heads in a spin of a penny. Which hypotheses do we want to test?


A$\begin{array}{l} H_0: p

eq 0.5 \\ H_a: p=0.5 \end{array}$

B$\begin{array}{l} H_0: p=0.5 \\ H_a: p

eq 0.5 \end{array}$

C$\begin{array}{l} H_0: p=0.5 \\ H_a: p>0.5 \end{array}$
D$\begin{array}{l} H_0: p=0.5 \\ H_a: p<0.5 \end{array}$


CCompute the $z$ test statistic.

$z=$

(Round to 2 places)


DIs the following statement True or False? There is strong evidence against equal probabilities.

Verified Answer :

  • The sample proportion of heads is \(\boxed{0.447}\).
  • The correct hypotheses are \(H_0: p = 0.5\) and \(H_a: p eq 0.5\).
  • The calculated \(z\)-test statistic is \(\boxed{-1.31}\).
  • The statement that there is strong evidence against equal probabilities is \(\boxed{\text{False}}\).

Explanation

  1. Calculate the sample proportion of heads

To analyze the results of the coin-spinning experiment, we first calculate the sample proportion of heads. With \(n = 150\) spins and \(X = 67\) heads, the sample proportion \(\hat{p}\) is:
\[\hat{p} = \frac{X}{n} = \frac{67}{150} \approx 0.44666...\]
Rounding to three decimal places as requested, we get \(\hat{p} = 0.447\).

  1. Define the null and alternative hypotheses

To determine if the probabilities of heads and tails are different, we perform a hypothesis test. The null hypothesis, denoted by \(H_0\), assumes there is no difference from the expected probability of a fair coin. The alternative hypothesis, denoted by \(H_a\), states the probability is not equal to that expected value.
Therefore, the correct hypotheses are:
\(H_0: p = 0.5\)
\(H_a: p eq 0.5\)

  1. Compute the z-test statistic

To determine if the observed proportion of \(0.447\) is significantly different from \(0.5\), we calculate the \(z\)-test statistic:
\[z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}\]
Plugging in the values \(\hat{p} = 0.44667\), \(p_0 = 0.5\), and \(n = 150\):
\[z = \frac{0.44667 - 0.5}{\sqrt{\frac{0.5 \times (1 - 0.5)}{150}}} = \frac{-0.05333}{\sqrt{\frac{0.25}{150}}} = \frac{-0.05333}{0.04082} \approx -1.306\]
Rounding to two decimal places, we find \(z = -1.31\).

  1. Determine the significance of the results

A \(z\)-statistic is considered to provide "strong evidence" against the null hypothesis if its absolute value is typically greater than \(1.96\) (corresponding to a standard significance level of \(\alpha = 0.05\)). Since our calculated value is \(|z| = 1.31\), which is less than \(1.96\), we do not have enough evidence to conclude that the probability of heads is different from \(0.5\). Therefore, the statement is False.

Examples

In statistics, researchers often use hypothesis testing to determine if an observed outcome in a sample significantly differs from a theoretical expectation. For instance, if a manufacturing process is calibrated to produce parts of a specific size, a random sample can be tested to check if the machine's calibration has drifted over time, using the same \(z\)-score method demonstrated here. Understanding these tests allows one to interpret whether variations in data are likely due to random chance or an actual underlying shift.

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