What is the formula for calculating the sample proportion, p hat?

A tech‐industry poll surveys $500$ software developers and finds $380$ of them prefer to write unit tests before coding new features. The team uses $300$ bootstrap replicates to estimate a $99\%$ confidence interval for the true proportion of all developers who follow this practice. From the bootstrap distribution, the $2.5$^th percentile is $0.730$ and the $97.5$^th percentile is $0.795$. Compute the normal‐approximation $99\%$ interval and compare it to the bootstrap result. 

A chi-square test produces a test statistic of ${\chi }^2=3.281$. Should you reject the null hypothesis based on the graph given below? Note: The purple area indicates the rejection region.

An experiment was conducted to determine whether providing candy to dining parties resulted in a higher tip percentage. The following data were collected on the tip percentages for two independent samples of dining parties:

Sample 1 (No Candy):

Sample size $\left(n{}_1\right)=40$

Mean tip percentage $\left({\bar{x}}_1\right)=15$

Standard deviation $\left(s_1\right)=4$

Sample 2 (Candy Given):

Sample size $\left(n_2\right)=45$

Mean tip percentage $({\bar{x}}_2=18$

Standard deviation $\left(s_2\right)=5$

Use a $0.05$ significance level to test the claim that giving candy results in greater tips.

A workforce analyst claims that more than half of remote employees believe their time management skills improved while working from home. In a random sample of $280$ remote employees, $158$ say their time management skills improved. At the $0.01$ significance level, is there enough evidence to support the analyst’s claim?

A study investigates the effectiveness of a new safety helmet for cyclists. Out of $1800$ cyclists not wearing the new helmet, $27$ suffered serious head injuries. Out of $5400$ cyclists wearing the new helmet, $9$ suffered serious head injuries. At the $0.05$ significance level, test the claim that the new helmet reduces the risk of serious head injury. What does the result suggest about the effectiveness of bicycle helmets?

A researcher claims that the median weekly hours worked by employees in a certain city is less than the national median of $38.5$ hours. The data collected from a random sample of employees in the city is quantitative, but the distribution is unknown. Which statistical test should be used to evaluate the researcher's claim about the median weekly hours?

A scientist wants to test whether a herbal supplement decreases reaction time. Reaction times are measured before and after the supplement for each participant. The scientist performs a left-tailed paired $t$-test at the $\alpha =0.05$ level. The test results lead to rejection of the null hypothesis. What is the correct interpretation of this result?

A study finds a correlation between ice cream sales and drowning incidents. What conclusion can be drawn?

A study finds a correlation coefficient of 0.78 between hours of sleep and productivity levels. What can be concluded about the relationship?

A retail chain predicts quarterly profit (in millions) using $y=12,300+4.50x_1-3.75x_2$, where $x_1$ is the number of new products launched and $x_2$ is the average marketing spend per product (in thousands). What is the predicted profit if $x_1=300$ and $x_2=800$?

How do residuals help in checking the assumptions of a regression model?

What is the effect of increasing the sample size on the variation of a dataset?

A regression equation for predicting the average number of hours spent exercising per week by adults ages $50$ to $59$ ($y$) from the average for ages $40$ to $49$ ($x$) is $\hat{y}=2.5+0.65x$. The sample size is $n=10$, $\bar{x}=3.7$, $S_{xx}=5.4$, and $s=0.5$. Construct a $90\%$ prediction interval for $y$ when $x=4.2$.

A candy company claims that their jelly beans are distributed equally among five colors. A sample of 100 jelly beans shows the following distribution: 18, 22, 20, 19, and 21. What is the expected frequency for each color if the claim is true?

Given a contingency table showing the number of students who prefer different study methods (group study, solo study) across two departments (Science, Arts), how would you calculate the marginal total for the Science department?

Why are degrees of freedom important in a Chi Square Test for Independence?

Which of the following is an assumption of the Chi-Square Test for Homogeneity?

What is the critical $F$-value for a two-tailed test with $\alpha =0.20$, numerator degrees of freedom $d.f.N=8$, and denominator degrees of freedom $d.f.D=16$?