A$.$ Hypothesis Testing $(70\%)$

Background: A technology company claims that the average download speed of their internet service is $100$ Mbps$.$ The company wants to test this claim with a sample of customers.

Task: You have been hired as a data analyst to help the technology company with hypothesis testing and confidence intervals.

$1.$ Formulate the null and alternative hypotheses for this situation.

$2.$ If a random sample of $50$ customers produces a sample mean download speed of $102$ Mbps with a sample standard deviation of $15$ Mbps$,$ calculate the test statistic using a z$-$value.

$3.$ Calculate the p$-$value for this test and interpret the results at a significance level of $0\mathrm{.}05.$

$4.$ Determine the critical value for a two$-$tailed test with a significance level of $0\mathrm{.}05.$

$5.$ Compare the calculated z$-$value from Task $2$ with the critical value obtained in Task $4.$ What is your conclusion?

$6.$ How would increasing the sample size from $50$ to $100$ affect your conclusion? Explain.

$7.$ Explain the concept of Type I and Type II errors in the context of hypothesis testing for the technology company's claim. Provide examples of each type of error and discuss their potential consequences.

$8.$ Provide two other examples of situations or industries where hypothesis testing and confidence intervals could be useful, and briefly explain why.

B$.$ Moving Average $(30\%)$

Imagine you are a data analyst for a music streaming service striving to improve user engagement and satisfaction. The company is interested in leveraging data$-$driven approaches to understand user behavior, enhance platform features, and increase user retention. Your task is to design a comprehensive data collection and analysis plan.

a$)$ Discuss the significance of various metrics and data sources in understanding user engagement and satisfaction within the music streaming platform. Provide examples of how each metric contributes to the overall analysis and suggest potential types of data sources that could be utilized. Additionally, refer to the table below, which displays sample user data including User ID$,$ Age, Gender, and Subscription type, and analyze how such data could be leveraged to gain insights into user behavior.

User Data Table

b$)$ Propose strategies for gathering user feedback to gain insights into user preferences and challenges. Describe different methods such as surveys, focus groups, or user interviews, and estimate the advantages and limitations of each approach. Consider the user data provided in the table and discuss how these feedback strategies could be applied to extract valuable insights from users.

c$)$ Formulate a hypothesis related to user engagement based on hypothetical data representing user activity on the music streaming platform over a one$-$month period. Outline a general method for testing this hypothesis using statistical analysis. Additionally, refer to the data trend plot below, illustrating the daily activity levels of users over the one$-$month period, and discuss how such trends could inform the hypothesis and its testing.