How can i do the following problem in r studio?

Question

Problem I.

It was reported on the local news that Gen Zers consider it rude if someone takes over 20 minutes to reply to a text message. To test this statement, you randomly select 30 Gen Zers to interview. You ask them, ”What is the longest length of time, in minutes, to reply to a text before you consider a person rude?” You calculated the following summary statistics: mean = 17.75 minutes and standard deviation = 3.0 minutes.

You set up the following pairs of hypotheses:

Ho : µ = 20 and Ha : µ > 20

1. What is the sample variance of the times?
2. Calculate the value of the test statistic.
3. Compute the p-value.
4. Test using α = 0.05. What is your conclusion about the statistical test and whether the length of time exceeds 20 minutes?
5. Compute a 95% confidence interval for µ to determine whether the news reported was correct. You may use the distribution multiplier of 2.045 (based on the t29 distribution). Interpret your results.

Answer 1

1. var = (std dev)^2. In your case, sample_var <- 3^2
2. t = (x - mu)/sigma. In your case, t <- (17.75 - 20)/(3/sqrt(30))
3. P-value is the probability of observing a value larger than your test statistic because your alternative hypothesis is mu > 20. In R, pnorm(t) = 2*10^-5 (approximately)
4. P-value < 0.05 so reject null
5. lower <- 17.75 - 2.045*3/sqrt(30) and upper <- 17.75 + 2.045*3/sqrt(30)

You could use a T-distribution by changing pnorm() to tnorm(), but the differences between a T-distribution with 29 degrees of freedom (30-1) and a normal distribution are negligible.