To determine the probability using a normal approximation we need to determine the z-score of the sample. To do that we use the following formula:
[tex]z=\frac{x-\mu}{\sigma}[/tex]Where:
[tex]\begin{gathered} z=\text{ z-score} \\ \mu=\text{ mean} \\ \sigma=\text{ standard deviation} \end{gathered}[/tex]First, we need to calculate the mean. To do that we use the following formula:
[tex]\mu=n\pi[/tex]Where
[tex]\begin{gathered} n=\text{ sample size} \\ \pi=\text{ failure rate in decimal form} \end{gathered}[/tex]To determine the decimal form of the failure rate we divide the percentage by 100, like this:
[tex]\pi=\frac{7.4}{100}=0.074[/tex]The sample size, in this case, is 295. Substituting we get:
[tex]\begin{gathered} \mu=(295)(0.074) \\ \mu=21.83 \end{gathered}[/tex]Now, we need to calculate the standard deviation. We do this using the following formula:
[tex]\sigma=\sqrt{n\pi(1-\pi)}[/tex]Substituting the values we get:
[tex]\sigma=\sqrt[]{(295)(0.074)(1-0.074)}[/tex]Solving the operations:
[tex]\sigma=4.49[/tex]Since we are approximating to binomial we need to use a continuity correction factor of 0.5 to the number of chips we are considering. Let "x" be the number of chips, we have that:
[tex]x=22-0.5=21.5[/tex]Now we substitute in the formula for the z-score:
[tex]z=\frac{21.5-21.83}{4.49}[/tex]Now, we solve the operations:
[tex]z=-0.073[/tex]Therefore, in the binomial distribution, we need to determine the area for z = -0.073. Therefore, the probability is:
[tex]P(x<22)=0.472[/tex]Therefore, the probability is 0.472