Question

Find the value of the test statistic z.The claim is that the proportion of peas with yellow pods is equal to 0.25 (or 25%). The sample statistics from one experiment include 430 peas with 126 of them having yellow pods.

Answer 1

You have to calculate the z-value or value of the test statistic z, for the proportion of yellow pods. The form of the z-statistic used to study the population proportion is:

$Z=\frac{\hat{p}-p}{\sqrt[]{\frac{p(1-p)}{n}}}\approx N(0,1)$

The population proportion of the yellow pods is p=0.25.

From a sample of 430 peas, 126 of them had yellow pods, using this information you have to calculate the sample proportion p-hat (^p)

The sample proportion is equal to the quotient between the number of successful outcomes "x" (number of yellow pods) and the total number of outcomes "n" (number of peas sampled)

$\begin{gathered} \hat{p}=\frac{x}{n} \\ \hat{p}=\frac{126}{430} \\ \hat{p}=0.29 \end{gathered}$

Once determined the sample proportion, you can calculate the z-value as follows:

$\begin{gathered} Z=\frac{0.29-0.25}{\sqrt[]{\frac{0.25(1-0.25)}{430}}} \\ Z=\frac{0.04}{\sqrt[]{\frac{0.25\cdot0.75}{430}}} \\ Z=\frac{0.04}{\sqrt[]{\frac{0.1875}{430}}} \\ Z=1.915\approx1.92 \end{gathered}$

The value of the test statistic is 1.92.

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The standard deviation of the distribution of the sample proportion:

To simplify the calculations you can calculate the standard deviation separately, the formula for this measure is the denominator of the formula of the Z-statistic:

$\sigma=\sqrt[]{\frac{p(1-p)}{n}}$

Using p=0.25

$\begin{gathered} \sigma=\sqrt[]{\frac{0.25(1-0.27)}{430}} \\ \sigma=\sqrt[]{\frac{0.25\cdot0.75}{430}} \\ \sigma=\sqrt[]{\frac{0.1875}{430}} \\ \sigma=\sqrt[]{\frac{3}{6880}} \\ \sigma=0.0208 \end{gathered}$

Then you can replace the result in the denominator of the formula to determine the Z-value:

$\begin{gathered} Z=\frac{\hat{p}-p}{\sqrt[]{\frac{p(1-p)}{n}}} \\ Z=\frac{0.29-0.25}{0.0208} \\ Z=\frac{0.04}{0.0208}\approx1.92 \end{gathered}$