91影视

Given in the question that, of the $98$teachers who responded, $23.5\%$said that they had one or more tattoos.

According to the information, we observed that

$n=98$

$\hat{p}=23.5\%=0.235$

For confidence level, $1-伪=0.95$, determine, $z_{伪/2}=z_{0.025}$

$z_{伪/2}=1.96$

The margin of error is

localid="1650685513171" $$\begin{gathered} E=z_{伪/2}路\sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \\ =1.96脳\sqrt{\frac{0.235(1-0.235)}{98}} \\ 鈮�0.084 \end{gathered}$$

The confidence interval is

localid="1650685516522" $$\begin{gathered} 0.151=0.235-0.084 \\ =\hat{p}-E<p<\hat{p}+E \\ =0.235+0.084 \\ =0.319 \end{gathered}$$

Of the $98$teachers who responded, $23.5\%$said that they had one or more tattoos.

We have to find the hypothesis

$H_0:p=0.14$

$H_0:p鈮�0.14$

Determine the value of the test-statistic:

localid="1650685524452" $$\begin{gathered} z=\frac{\hat{p}-p_0}{\sqrt{\frac{p_0\left(1-p_0\right)}{n}}} \\ =\frac{0.235-0.14}{\sqrt{\frac{0.14(1-0.14)}{98}}} \\ 鈮�2.71 \end{gathered}$$

The localid="1650685529133" $P$-value is the chance of getting the test statistic's result, or a number that is more severe. Calculate the localid="1650685533236" $P$-value in table localid="1650685536989" $A$as follows:

localid="1650685554330" $$\begin{gathered} P=P(Z<-2.71orZ>2.71) \\ =2脳P(Z<-2.71) \\ =2脳0.0034 \\ =0.0068 \end{gathered}$$

Reject the null hypothesis if the P-value is less than the significance level:

$P<0.05鈬�Reject{H}_0$

Of the $98$teachers who responded, $23.5\%$said that they had one or more tattoos.

From the given values

The sample proportion is calculated by dividing the number of successes by the sample size:

If both professors indicated they had one or more tattoos, the following would happen:

$$\begin{gathered} \hat{p}=\frac{x}{n} \\ =\frac{23.5\%脳98+2}{100} \\ 鈮�\frac{25}{100} \\ =0.25 \end{gathered}$$

If both teachers indicated they didn't have any tattoos, then:

$$\begin{gathered} \hat{p}=\frac{x}{n} \\ =\frac{23.5\%脳98+0}{100} \\ 鈮�\frac{23}{100} \\ =0.23 \end{gathered}$$

Let's determine the hypothesis

$H_0:p=0.14$

$H_0:p鈮�0.14$

Compute the test statistic's value:

localid="1650685575009" $$\begin{gathered} P=P(Z<-3.17orZ>3.17) \\ =2脳P(Z<-3.17) \\ =2脳0.0008 \\ =0.0016 \end{gathered}$$

localid="1650685581138" $$\begin{gathered} P=P(Z<-2.59orZ>2.59) \\ =2脳P(Z<-2.59) \\ =2脳0.0048 \\ =0.0096 \end{gathered}$$

Reject the null hypothesis if thelocalid="1650685612366" $P$-value is less than the significance level.

$P<0.05鈬�Reject{H}_0$