A Washington, D.C., "think tank" announces the typical...

A Washington, D.C., "think tank" announces the typical teenager sent 67 text messages per day in 2017 . To update that estimate, you phone a sample of 12 teenagers and ask them how many text messages they sent the previous day. Their responses were: $\begin{array}{llllllllll}51 & 175 & 47 & 49 & 44 & 54 & 145 & 203 & 21 & 59 & 42 & 100\end{array}$ At the .05 level, can you conclude that the mean number is greater than $67 ?$ Estimate the $p$ -value and describe what it tells you.

A Washington, D.C., "think tank" announces the typical teenager sent 67 text messages per day in 2017 . To update that estimate, you phone a sample of 12 teenagers and ask them how many text messages they sent the previous day. Their responses were:
$\begin{array}{llllllllll}51 & 175 & 47 & 49 & 44 & 54 & 145 & 203 & 21 & 59 & 42 & 100\end{array}$
At the .05 level, can you conclude that the mean number is greater than $67 ?$ Estimate the $p$ -value and describe what it tells you.

Key Concepts

Significance Level

The significance level, denoted as ?, is a threshold set before testing that determines the maximum acceptable probability of mistakenly rejecting the null hypothesis (a Type I error). If the p-value is less than ?, the result is considered statistically significant, leading to the rejection of the null hypothesis.

p-value

The p-value is the probability of obtaining a test statistic at least as extreme as the one observed, assuming that the null hypothesis is true. It provides a measure of the strength of evidence against the null hypothesis: a smaller p-value indicates stronger evidence for rejecting the null hypothesis.

Test Statistic

The test statistic is a standardized value derived from sample data used to assess the evidence against the null hypothesis. In a one-sample t-test, it is calculated by subtracting the hypothesized mean from the sample mean and then dividing by the standard error of the mean, which is estimated from the sample standard deviation and size.

Hypothesis Testing

Hypothesis testing is a systematic approach used to decide whether there is enough statistical evidence in a sample of data to infer that a certain condition holds for the entire population. It involves setting up a null hypothesis, which represents no effect or a baseline state, and an alternative hypothesis, which represents the claim or condition being tested.

One-Sample t-Test

The one-sample t-test is used to compare the mean of a single sample to a known or hypothesized population mean when the population standard deviation is unknown and the sample size is relatively small. This test computes a t-statistic by standardizing the difference between the sample mean and the hypothesized mean, taking into account the sample’s variability and size.

Transcript

00:01 All right, so there's a group in dc that ounces as a typical teenager sent 67 text messages per day in 2017.

00:11 So to update the estimate, we sampled 12 teenagers, and that's the only text messages they sent the previous day.

00:20 So we're going to assume it's normally distributed.

00:23 And these are their responses of how many they sent in here.

00:30 We're going to need the mean and standard deviation.

00:32 So let's go ahead and do that.

00:34 Let's see the average of the data.

00:36 I say average, that's the function i'm using.

00:41 I can just copy and paste and change the formula to sample standard deviation.

00:48 Good, they're good, done.

00:51 Let's see, now let's state our hypotheses, h0 is that, let's see, the new is less than, because let's see, we're asking, can we conclude the mean is greater than 67? so we're going to say less than 67 for the null.

01:15 The alternative is that no, mu is greater than 67.

01:21 And this tells us our decision for our critical value.

01:25 So this is going to say whatever the critical value is, the t statistic needs to be greater than it.

01:31 So let's see 12, so 11 degrees of freedom at the 0 .05 significance level.

01:37 0 .05, it's one -tailed test, 11 degrees of freedom, right there.

01:45 Now the t statistic, we have the values, we can do it...

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