Statistic Assignment

APPENDIX

#Question 1

# calculate the probability that more than 5 resident loads arrive in the next 5 minutes
lambda <- 50 # Average rate
time_interval <- 0.0833 # 5 minutes in hours

# Calculate the sum of probabilities for k = 0 to 5
prob_sum <- sum(dpois(0:5, lambda * time_interval))

# Probability of more than 5 arrivals
prob_more_than_5 <- 1 - prob_sum
prob_more_than_5

# Question 2

# calculate the confidence interval
# Given values
sample_size <- 130
bamboo_loads <- 18
confidence_level <- 0.95

# Calculate the sample proportion
sample_proportion <- bamboo_loads / sample_size

# Calculate the standard error
standard_error <- sqrt((sample_proportion * (1 - sample_proportion)) / sample_size)

# Calculate the critical value for the given confidence level
critical_value <- qnorm((1 + confidence_level) / 2)

# Calculate the margin of error
margin_of_error <- critical_value * standard_error

# Calculate the confidence interval
lower_bound <- sample_proportion - margin_of_error
upper_bound <- sample_proportion + margin_of_error
# Print the confidence interval
cat("95% Confidence Interval:", lower_bound, "to", upper_bound)

# Question 3

# hi-squared test for variance
# Given data
data <- c(63.9, 78.5, 58.1, 109.4, 87.2, 76.3, 88.5, 66.4, 85.8, 102.1, 71.9, 89.9, 86.7, 68.3, 93.5)
# Degrees of freedom is the sample size minus 1
df <- length(data) - 1
# Calculate the sample variance
sample_variance <- var(data)
# Null hypothesis: Variance = 100
null_variance <- 100
# Calculate the test statistic
test_statistic <- (df * sample_variance) / null_variance
# Calculate the critical value for the Chi-squared distribution
alpha <- 0.05
critical_value <- qchisq(1 - alpha, df)
# Compare the test statistic with the critical value
if (test_statistic > critical_value) {
cat("Reject the null hypothesis. Variance is greater than 100.")
} else {
cat("Fail to reject the null hypothesis. Variance is not significantly different from 100.")
}
# Print test statistic and critical value
cat("nTest statistic:", test_statistic)
cat("nCritical value:", critical_value)

# Question 4

# Given values
sample_size <- 10
sample_mean <- 56
population_mean <- 50
population_stddev <- 10
# Calculate the standard error
standard_error <- population_stddev / sqrt(sample_size)
# Calculate the t-statistic
t_statistic <- (sample_mean - population_mean) / standard_error
# Degrees of freedom
df <- sample_size - 1
# Calculate the critical value for the t-distribution
alpha <- 0.05
critical_value <- qt(1 - alpha, df)
# Compare the t-statistic with the critical value
if (t_statistic > critical_value) {
cat("Reject the null hypothesis. Training leads to an increase in MVPA minutes.")
} else {
cat("Fail to reject the null hypothesis. Training does not lead to a significant increase.")
}
# Print t-statistic and critical value
cat("nT-statistic:", t_statistic)
cat("nCritical value:", critical_value)

# Question 5

# Given values for before installation
mean_before <- 12.0
variance_before <- 100.17
sample_size_before <- 8
# Given values for after installation
mean_after <- 10.2
variance_after <- 96.73
sample_size_after <- 9
# Calculate the pooled standard error
pooled_std_error <- sqrt((variance_before / sample_size_before) + (variance_after / sample_size_after))
# Calculate the t-statistic
t_statistic <- (mean_before - mean_after) / pooled_std_error
# Degrees of freedom
df <- sample_size_before + sample_size_after - 2
# Calculate the critical value for the t-distribution
alpha <- 0.05
critical_value <- qt(1 - alpha, df)
# Compare the t-statistic with the critical value
if (t_statistic > critical_value) {
cat("Reject the null hypothesis. The filtering device has reduced the percentage of impurity significantly.")
} else {
cat("Fail to reject the null hypothesis. There is no significant reduction in the percentage of impurity.")
}
# Print t-statistic and critical value
cat("nT-statistic:", t_statistic)
cat("nCritical value:", critical_value)

# Question 6

# Given values for before policy change
assignments_before <- 250
not_turned_in_before <- 42
# Given values for after policy change
assignments_after <- 210
not_turned_in_after <- 15
# Calculate sample proportions
p_hat_before <- not_turned_in_before / assignments_before
p_hat_after <- not_turned_in_after / assignments_after
# Calculate the pooled sample proportion
p_pooled <- (not_turned_in_before + not_turned_in_after) / (assignments_before + assignments_after)
# Calculate the standard error
se <- sqrt(p_pooled * (1 - p_pooled) * ((1 / assignments_before) + (1 / assignments_after)))
# Calculate the test statistic
z_statistic <- (p_hat_before - p_hat_after) / se
# Calculate the p-value
p_value <- 1 - pnorm(z_statistic)
p_value

# Question 7

# Given data
data <- matrix(c(31, 68, 20, 42, 36, 20, 50, 30, 25, 30, 35, 23), nrow = 4, byrow = TRUE)
rownames(data) <- c("scratches", "structure", "hardware", "finish")
colnames(data) <- c("A", "B", "C")
# Perform chi-squared test
chi_square_result <- chisq.test(data)
# Print the p-value
p_value <- chi_square_result$p.value
p_value

# Question 8

# Given parameters
mu_X <- 4
mu_Y <- 8
sigma_X <- sqrt(5)
sigma_Y <- sqrt(9)
rho <- 0.3
y_condition <- 3
x_condition <- 6
# Calculate the conditional mean and variance of X given Y = 3
mu_X_given_Y <- mu_X + rho * (sigma_X / sigma_Y) * (y_condition - mu_Y)
sigma_X_given_Y <- sigma_X * sqrt(1 - rho^2)
# Calculate the z-score
z_score <- (x_condition - mu_X_given_Y) / sigma_X_given_Y
# Calculate the conditional probability
conditional_probability <- pnorm(z_score)
conditional_probability

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