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The sample of 26 ratings is observed such that each rating varies from 1 to 10. The confidence level is 99%.

The necessary conditions for using any sample data to construct confidence intervals are as follows.

The sample is collected from the population of females that satisfies the condition of a simple random sampling. As the sample size is 26, which is less than 30, the condition for normality will only be satisfied if the data follows a normal distribution.

Assume that the requirements hold true.

The degree of freedom is computed as follows.

Substitute 26 for in the above formula and simplify.

$$\begin{gathered} df=n-1 \\ =26-1 \\ =25 \end{gathered}$$

The level of significance is 0.01 for the confidence level of 0.99.

.$$\begin{gathered} 伪=1-0.99 \\ =0.01 \end{gathered}$$

Use the t-distribution table to obtain the critical value when $伪=0.01$and $df=25$.

The value corresponding to row 25 and column 0.01 (two-tailed) is obtained as $t_{\frac{0.01}{2}}=2.787$.

Let $X$be the random variable that denotes the rating of females.

The sample mean can be obtained using the formula $\stackrel{炉}{x}=\frac{1}{26}\underset{i=1}{\overset{26}{鈭�}}{x}_i$, where$x_i$represents the data points in a sample.

Compute the sample mean.

$$\begin{gathered} \stackrel{炉}{x}=\frac{\left(5+8+3...+7\right)}{26} \\ =\frac{171}{26} \\ =6.5769 \end{gathered}$$

Calculate the sample variance using the formula $s^2=\frac{1}{26-1}\underset{i=1}{\overset{26}{鈭�}}{\left(x_i-\stackrel{炉}{x}\right)}^2$.

Subsitute $\underset{i=1}{\overset{26}{鈭�}}{\left(x_i-\stackrel{炉}{x}\right)}^2=88.16$in the formula $s^2=\frac{1}{25-1}\underset{i=1}{\overset{26}{鈭�}}{\left(x_i-\stackrel{炉}{x}\right)}^2$. So,

$$\begin{gathered} s^2=\frac{1}{25}脳88.16 \\ =\frac{88.16}{25} \\ =3.526 \end{gathered}$$

The square root of the sample variance is equal to the sample standard deviation. Thus, the sample standard deviation is given as follows.

$$\begin{gathered} s=\sqrt{3.526} \\ =1.8798 \end{gathered}$$

The margin of error is given by the formula $E=t_{\frac{伪}{2}}脳\frac{s}{\sqrt{n}}$.Substitute the respective value obtained from above in the equation and simplify to compute the margin of error. So,

$$\begin{gathered} E=2.787脳\frac{1.8798}{\sqrt{26}} \\ =1.0276 \end{gathered}$$

The 99% confidence interval is given as follows.

$$\begin{gathered} \stackrel{炉}{x}-E<渭<\stackrel{炉}{x}-E \\ 6.5769-1.0276<渭<6.5769+1.0276 \\ 5.5493<渭<7.6046 \end{gathered}$$

Thus, the 99% confidence level for the mean attractiveness of males is between 5.5 and 7.6.

The mean attractiveness of male dates is based on the ratings of female subjects, which cannot be used to infer about the population of all adult males.

Construct the normal probability plot to check if the sample data follows a normal distributionto check the applicability of the confidence interval.

1. Draw two axes for the observed values (X) and z-scores(Z).
2. Mark the observations corresponding to the z-scores.
3. Observe if the marks follow a straight-line pattern.

Thus, there exists an outlier, and the values do not follow a straight-line pattern. Thus, it can be inferred that the confidence interval is not useful for estimating the mean.